Source code for sympy.matrices.matrices

from __future__ import print_function, division

import collections
from sympy.core.add import Add
from sympy.core.basic import Basic, Atom
from sympy.core.expr import Expr
from sympy.core.power import Pow
from sympy.core.symbol import Symbol, Dummy, symbols
from sympy.core.numbers import Integer, ilcm, Float
from sympy.core.singleton import S
from sympy.core.sympify import sympify
from sympy.core.compatibility import is_sequence, default_sort_key, range, \
    NotIterable

from sympy.polys import PurePoly, roots, cancel, gcd
from sympy.simplify import simplify as _simplify, signsimp, nsimplify
from sympy.utilities.iterables import flatten, numbered_symbols
from sympy.functions.elementary.miscellaneous import sqrt, Max, Min
from sympy.functions import Abs, exp, factorial
from sympy.printing import sstr
from sympy.core.compatibility import reduce, as_int, string_types
from sympy.assumptions.refine import refine
from sympy.core.decorators import call_highest_priority

from types import FunctionType

from .common import (a2idx, classof, MatrixError, ShapeError,
        NonSquareMatrixError, MatrixCommon)


def _iszero(x):
    """Returns True if x is zero."""
    try:
        return x.is_zero
    except AttributeError:
        return None


class DeferredVector(Symbol, NotIterable):
    """A vector whose components are deferred (e.g. for use with lambdify)

    Examples
    ========

    >>> from sympy import DeferredVector, lambdify
    >>> X = DeferredVector( 'X' )
    >>> X
    X
    >>> expr = (X[0] + 2, X[2] + 3)
    >>> func = lambdify( X, expr)
    >>> func( [1, 2, 3] )
    (3, 6)
    """

    def __getitem__(self, i):
        if i == -0:
            i = 0
        if i < 0:
            raise IndexError('DeferredVector index out of range')
        component_name = '%s[%d]' % (self.name, i)
        return Symbol(component_name)

    def __str__(self):
        return sstr(self)

    def __repr__(self):
        return "DeferredVector('%s')" % self.name


class MatrixDeterminant(MatrixCommon):
    """Provides basic matrix determinant operations.
    Should not be instantiated directly."""

    def _eval_berkowitz_toeplitz_matrix(self):
        """Return (A,T) where T the Toeplitz matrix used in the Berkowitz algorithm
        corresponding to `self` and A is the first principal submatrix."""

        # the 0 x 0 case is trivial
        if self.rows == 0 and self.cols == 0:
            return self._new(1,1, [S.One])

        #
        # Partition self = [ a_11  R ]
        #                  [ C     A ]
        #

        a, R = self[0,0],   self[0, 1:]
        C, A = self[1:, 0], self[1:,1:]

        #
        # The Toeplitz matrix looks like
        #
        #  [ 1                                     ]
        #  [ -a         1                          ]
        #  [ -RC       -a        1                 ]
        #  [ -RAC     -RC       -a       1         ]
        #  [ -RA**2C -RAC      -RC      -a       1 ]
        #  etc.

        # Compute the diagonal entries.
        # Because multiplying matrix times vector is so much
        # more efficient than matrix times matrix, recursively
        # compute -R * A**n * C.
        diags = [C]
        for i in range(self.rows - 2):
            diags.append(A * diags[i])
        diags = [(-R*d)[0, 0] for d in diags]
        diags = [S.One, -a] + diags

        def entry(i,j):
            if j > i:
                return S.Zero
            return diags[i - j]

        toeplitz = self._new(self.cols + 1, self.rows, entry)
        return (A, toeplitz)

    def _eval_berkowitz_vector(self):
        """ Run the Berkowitz algorithm and return a vector whose entries
            are the coefficients of the characteristic polynomial of `self`.

            Given N x N matrix, efficiently compute
            coefficients of characteristic polynomials of 'self'
            without division in the ground domain.

            This method is particularly useful for computing determinant,
            principal minors and characteristic polynomial when 'self'
            has complicated coefficients e.g. polynomials. Semi-direct
            usage of this algorithm is also important in computing
            efficiently sub-resultant PRS.

            Assuming that M is a square matrix of dimension N x N and
            I is N x N identity matrix, then the Berkowitz vector is
            an N x 1 vector whose entries are coefficients of the
            polynomial

                           charpoly(M) = det(t*I - M)

            As a consequence, all polynomials generated by Berkowitz
            algorithm are monic.

           For more information on the implemented algorithm refer to:

           [1] S.J. Berkowitz, On computing the determinant in small
               parallel time using a small number of processors, ACM,
               Information Processing Letters 18, 1984, pp. 147-150

           [2] M. Keber, Division-Free computation of sub-resultants
               using Bezout matrices, Tech. Report MPI-I-2006-1-006,
               Saarbrucken, 2006
        """

        # handle the trivial cases
        if self.rows == 0 and self.cols == 0:
            return self._new(1, 1, [S.One])
        elif self.rows == 1 and self.cols == 1:
            return self._new(2, 1, [S.One, -self[0,0]])

        submat, toeplitz = self._eval_berkowitz_toeplitz_matrix()
        return toeplitz * submat._eval_berkowitz_vector()

    def _eval_det_bareiss(self):
        """Compute matrix determinant using Bareiss' fraction-free
        algorithm which is an extension of the well known Gaussian
        elimination method. This approach is best suited for dense
        symbolic matrices and will result in a determinant with
        minimal number of fractions. It means that less term
        rewriting is needed on resulting formulae.

        TODO: Implement algorithm for sparse matrices (SFF),
        http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.
        """

        # XXX included as a workaround for issue #12362.  Should use `_find_reasonable_pivot` instead
        def _find_pivot(l):
            for pos,val in enumerate(l):
                if val:
                    return (pos, val, None, None)
            return (None, None, None, None)


        # Recursively implimented Bareiss' algorithm as per Deanna Richelle Leggett's
        # thesis http://www.math.usm.edu/perry/Research/Thesis_DRL.pdf
        def bareiss(mat, cumm=1):
            if mat.rows == 0:
                return S.One
            elif mat.rows == 1:
                return mat[0, 0]

            # find a pivot and extract the remaining matrix
            # XXX should use `_find_reasonable_pivot`.  Blocked by issue #12362
            pivot_pos, pivot_val, _, _ = _find_pivot(mat[:, 0])
            if pivot_pos == None:
                return S.Zero

            # if we have a valid pivot, we'll do a "row swap", so keep the
            # sign of the det
            sign = (-1) ** (pivot_pos % 2)

            # we want every row but the pivot row and every column
            rows = list(i for i in range(mat.rows) if i != pivot_pos)
            cols = list(range(mat.cols))
            tmp_mat = mat.extract(rows, cols)

            def entry(i, j):
                ret = (pivot_val*tmp_mat[i, j + 1] - mat[pivot_pos, j + 1]*tmp_mat[i, 0]) / cumm
                if not ret.is_Atom:
                    cancel(ret)
                return ret

            return sign*bareiss(self._new(mat.rows - 1, mat.cols - 1, entry), pivot_val)

        return cancel(bareiss(self))

    def _eval_det_berkowitz(self):
        """ Use the Berkowitz algorithm to compute the determinant."""
        berk_vector = self._eval_berkowitz_vector()
        return (-1)**(len(berk_vector) - 1) * berk_vector[-1]

    def _eval_det_lu(self, iszerofunc=_iszero, simpfunc=None):
        """ Computes the determinant of a matrix from its LU decomposition.
        This function uses the LU decomposition computed by
        LUDecomposition_Simple().

        The keyword arguments iszerofunc and simpfunc are passed to
        LUDecomposition_Simple().
        iszerofunc is a callable that returns a boolean indicating if its
        input is zero, or None if it cannot make the determination.
        simpfunc is a callable that simplifies its input.
        The default is simpfunc=None, which indicate that the pivot search
        algorithm should not attempt to simplify any candidate pivots.
        If simpfunc fails to simplify its input, then it must return its input
        instead of a copy."""

        if self.rows == 0:
            return S.One
            # sympy/matrices/tests/test_matrices.py contains a test that
            # suggests that the determinant of a 0 x 0 matrix is one, by
            # convention.

        lu, row_swaps = self.LUdecomposition_Simple(iszerofunc=iszerofunc, simpfunc=None)
        # P*A = L*U => det(A) = det(L)*det(U)/det(P) = det(P)*det(U).
        # Lower triangular factor L encoded in lu has unit diagonal => det(L) = 1.
        # P is a permutation matrix => det(P) in {-1, 1} => 1/det(P) = det(P).
        # LUdecomposition_Simple() returns a list of row exchange index pairs, rather
        # than a permutation matrix, but det(P) = (-1)**len(row_swaps).

        # Avoid forming the potentially time consuming  product of U's diagonal entries
        # if the product is zero.
        # Bottom right entry of U is 0 => det(A) = 0.
        # It may be impossible to determine if this entry of U is zero when it is symbolic.
        if iszerofunc(lu[lu.rows-1, lu.rows-1]):
            return S.Zero

        # Compute det(P)
        det = -S.One if len(row_swaps)%2 else S.One

        # Compute det(U) by calculating the product of U's diagonal entries.
        # The upper triangular portion of lu is the upper triangular portion of the
        # U factor in the LU decomposition.
        for k in range(lu.rows):
            det *= lu[k, k]

        # return det(P)*det(U)
        return det

    def _eval_determinant(self):
        """Assumed to exist by matrix expressions; If we subclass
        MatrixDeterminant, we can fully evaluate determinants."""
        return self.det()

    def adjugate(self, method="berkowitz"):
        """Returns the adjugate, or classical adjoint, of
        a matrix.  That is, the transpose of the matrix of cofactors.


        http://en.wikipedia.org/wiki/Adjugate

        See Also
        ========

        cofactor_matrix
        transpose
        """
        return self.cofactor_matrix(method).transpose()

    def charpoly(self, x=Dummy('lambda'), simplify=_simplify):
        """Computes characteristic polynomial det(x*I - self) where I is
        the identity matrix.

        A PurePoly is returned, so using different variables for ``x`` does
        not affect the comparison or the polynomials:

        Examples
        ========

        >>> from sympy import Matrix
        >>> from sympy.abc import x, y
        >>> A = Matrix([[1, 3], [2, 0]])
        >>> A.charpoly(x) == A.charpoly(y)
        True

        Specifying ``x`` is optional; a Dummy with name ``lambda`` is used by
        default (which looks good when pretty-printed in unicode):

        >>> A.charpoly().as_expr()
        _lambda**2 - _lambda - 6

        No test is done to see that ``x`` doesn't clash with an existing
        symbol, so using the default (``lambda``) or your own Dummy symbol is
        the safest option:

        >>> A = Matrix([[1, 2], [x, 0]])
        >>> A.charpoly().as_expr()
        _lambda**2 - _lambda - 2*x
        >>> A.charpoly(x).as_expr()
        x**2 - 3*x

        Notes
        =====

        The Samuelson-Berkowitz algorithm is used to compute
        the characteristic polynomial efficiently and without any
        division operations.  Thus the characteristic polynomial over any
        commutative ring without zero divisors can be computed.

        See Also
        ========

        det
        """

        if self.rows != self.cols:
            raise NonSquareMatrixError()

        berk_vector = self._eval_berkowitz_vector()
        return PurePoly([simplify(a) for a in berk_vector], x)

    def cofactor(self, i, j, method="berkowitz"):
        """Calculate the cofactor of an element.

        See Also
        ========

        cofactor_matrix
        minor
        minor_submatrix
        """

        if self.rows != self.cols or self.rows < 1:
            raise NonSquareMatrixError()

        return (-1)**((i + j) % 2) * self.minor(i, j, method)

    def cofactor_matrix(self, method="berkowitz"):
        """Return a matrix containing the cofactor of each element.

        See Also
        ========

        cofactor
        minor
        minor_submatrix
        adjugate
        """

        if self.rows != self.cols or self.rows < 1:
            raise NonSquareMatrixError()

        return self._new(self.rows, self.cols,
                         lambda i, j: self.cofactor(i, j, method))

    def det(self, method="bareiss"):
        """Computes the determinant of a matrix.  If the matrix
        is at most 3x3, a hard-coded formula is used.
        Otherwise, the determinant using the method `method`.


        Possible values for "method":
          bareis
          berkowitz
          lu
        """

        # sanitize `method`
        method = method.lower()
        if method == "bareis":
            method = "bareiss"
        if method == "det_lu":
            method = "lu"
        if method not in ("bareiss", "berkowitz", "lu"):
            raise ValueError("Determinant method '%s' unrecognized" % method)

        # if methods were made internal and all determinant calculations
        # passed through here, then these lines could be factored out of
        # the method routines
        if self.rows != self.cols:
            raise NonSquareMatrixError()

        n = self.rows
        if n == 0:
            return S.One
        elif n == 1:
            return self[0,0]
        elif n == 2:
            return self[0, 0] * self[1, 1] - self[0, 1] * self[1, 0]
        elif n == 3:
            return  (self[0, 0] * self[1, 1] * self[2, 2]
                   + self[0, 1] * self[1, 2] * self[2, 0]
                   + self[0, 2] * self[1, 0] * self[2, 1]
                   - self[0, 2] * self[1, 1] * self[2, 0]
                   - self[0, 0] * self[1, 2] * self[2, 1]
                   - self[0, 1] * self[1, 0] * self[2, 2])

        if method == "bareiss":
            return self._eval_det_bareiss()
        elif method == "berkowitz":
            return self._eval_det_berkowitz()
        elif method == "lu":
            return self._eval_det_lu()

    def minor(self, i, j, method="berkowitz"):
        """Return the (i,j) minor of `self`.  That is,
        return the determinant of the matrix obtained by deleting
        the `i`th row and `j`th column from `self`.

        See Also
        ========

        minor_submatrix
        cofactor
        det
        """

        if self.rows != self.cols or self.rows < 1:
            raise NonSquareMatrixError()

        return self.minor_submatrix(i, j).det(method=method)

    def minor_submatrix(self, i, j):
        """Return the submatrix obtained by removing the `i`th row
        and `j`th column from `self`.

        See Also
        ========

        minor
        cofactor
        """

        if i < 0:
            i += self.rows
        if j < 0:
            j += self.cols

        if not 0 <= i < self.rows or not 0 <= j < self.cols:
            raise ValueError("`i` and `j` must satisfy 0 <= i < `self.rows` "
                             "(%d)" % self.rows + "and 0 <= j < `self.cols` (%d)." % self.cols)

        rows = [a for a in range(self.rows) if a != i]
        cols = [a for a in range(self.cols) if a != j]
        return self.extract(rows, cols)


class MatrixReductions(MatrixDeterminant):
    """Provides basic matrix row/column operations.
    Should not be instantiated directly."""

    def _eval_col_op_swap(self, col1, col2):
        def entry(i, j):
            if j == col1:
                return self[i, col2]
            elif j == col2:
                return self[i, col1]
            return self[i, j]
        return self._new(self.rows, self.cols, entry)

    def _eval_col_op_multiply_col_by_const(self, col, k):
        def entry(i, j):
            if j == col:
                return k * self[i, j]
            return self[i, j]
        return self._new(self.rows, self.cols, entry)

    def _eval_col_op_add_multiple_to_other_col(self, col, k, col2):
        def entry(i, j):
            if j == col:
                return self[i, j] + k * self[i, col2]
            return self[i, j]
        return self._new(self.rows, self.cols, entry)

    def _eval_row_op_swap(self, row1, row2):
        def entry(i, j):
            if i == row1:
                return self[row2, j]
            elif i == row2:
                return self[row1, j]
            return self[i, j]
        return self._new(self.rows, self.cols, entry)

    def _eval_row_op_multiply_row_by_const(self, row, k):
        def entry(i, j):
            if i == row:
                return k * self[i, j]
            return self[i, j]
        return self._new(self.rows, self.cols, entry)

    def _eval_row_op_add_multiple_to_other_row(self, row, k, row2):
        def entry(i, j):
            if i == row:
                return self[i, j] + k * self[row2, j]
            return self[i, j]
        return self._new(self.rows, self.cols, entry)

    def _eval_echelon_form(self, iszerofunc, simpfunc):
        """Returns (mat, swaps) where `mat` is a row-equivalent matrix
        in echelon form and `swaps` is a list of row-swaps performed."""
        reduced, pivot_cols, swaps = self._row_reduce(iszerofunc, simpfunc,
                                                      normalize_last=True,
                                                      normalize=False,
                                                      zero_above=False)
        return reduced, pivot_cols, swaps

    def _eval_is_echelon(self, iszerofunc):
        if self.rows <= 0 or self.cols <= 0:
            return True
        zeros_below = all(iszerofunc(t) for t in self[1:, 0])
        if iszerofunc(self[0, 0]):
            return zeros_below and self[:, 1:]._eval_is_echelon(iszerofunc)
        return zeros_below and self[1:, 1:]._eval_is_echelon(iszerofunc)

    def _eval_rref(self, iszerofunc, simpfunc, normalize_last=True):
        reduced, pivot_cols, swaps = self._row_reduce(iszerofunc, simpfunc,
                                                      normalize_last, normalize=True,
                                                      zero_above=True)
        return reduced, pivot_cols

    def _normalize_op_args(self, op, col, k, col1, col2, error_str="col"):
        """Validate the arguments for a row/column operation.  `error_str`
        can be one of "row" or "col" depending on the arguments being parsed."""
        if op not in ["n->kn", "n<->m", "n->n+km"]:
            raise ValueError("Unknown {} operation '{}'. Valid col operations "
                             "are 'n->kn', 'n<->m', 'n->n+km'".format(error_str, op))

        # normalize and validate the arguments
        if op == "n->kn":
            col = col if col is not None else col1
            if col is None or k is None:
                raise ValueError("For a {0} operation 'n->kn' you must provide the "
                                 "kwargs `{0}` and `k`".format(error_str))
            if not 0 <= col <= self.cols:
                raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col))

        if op == "n<->m":
            # we need two cols to swap. It doesn't matter
            # how they were specified, so gather them together and
            # remove `None`
            cols = set((col, k, col1, col2)).difference([None])
            if len(cols) > 2:
                # maybe the user left `k` by mistake?
                cols = set((col, col1, col2)).difference([None])
            if len(cols) != 2:
                raise ValueError("For a {0} operation 'n<->m' you must provide the "
                                 "kwargs `{0}1` and `{0}2`".format(error_str))
            col1, col2 = cols
            if not 0 <= col1 <= self.cols:
                raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col1))
            if not 0 <= col2 <= self.cols:
                raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col2))

        if op == "n->n+km":
            col = col1 if col is None else col
            col2 = col1 if col2 is None else col2
            if col is None or col2 is None or k is None:
                raise ValueError("For a {0} operation 'n->n+km' you must provide the "
                                 "kwargs `{0}`, `k`, and `{0}2`".format(error_str))
            if col == col2:
                raise ValueError("For a {0} operation 'n->n+km' `{0}` and `{0}2` must "
                                 "be different.".format(error_str))
            if not 0 <= col <= self.cols:
                raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col))
            if not 0 <= col2 <= self.cols:
                raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col2))

        return op, col, k, col1, col2

    def _permute_complexity_right(self, iszerofunc):
        """Permute columns with complicated elements as
        far right as they can go.  Since the `sympy` row reduction
        algorithms start on the left, having complexity right-shifted
        speeds things up.

        Returns a tuple (mat, perm) where perm is a permutation
        of the columns to perform to shift the complex columns right, and mat
        is the permuted matrix."""

        def complexity(i):
            # the complexity of a column will be judged by how many
            # element's zero-ness cannot be determined
            return sum(1 if iszerofunc(e) is None else 0 for e in self[:, i])
        complex = [(complexity(i), i) for i in range(self.cols)]
        perm = [j for (i, j) in sorted(complex)]

        return (self.permute(perm, orientation='cols'), perm)

    def _row_reduce(self, iszerofunc, simpfunc, normalize_last=True,
                    normalize=True, zero_above=True):
        """Row reduce `self` and return a tuple (rref_matrix,
        pivot_cols, swaps) where pivot_cols are the pivot columns
        and swaps are any row swaps that were used in the process
        of row reduction.

        Parameters
        ==========

        iszerofunc : determines if an entry can be used as a pivot
        simpfunc : used to simplify elements and test if they are
            zero if `iszerofunc` returns `None`
        normalize_last : indicates where all row reduction should
            happen in a fraction-free manner and then the rows are
            normalized (so that the pivots are 1), or whether
            rows should be normalized along the way (like the naive
            row reduction algorithm)
        normalize : whether pivot rows should be normalized so that
            the pivot value is 1
        zero_above : whether entries above the pivot should be zeroed.
            If `zero_above=False`, an echelon matrix will be returned.
        """
        rows, cols = self.rows, self.cols
        mat = list(self)
        def get_col(i):
            return mat[i::cols]

        def row_swap(i, j):
            mat[i*cols:(i + 1)*cols], mat[j*cols:(j + 1)*cols] = \
                mat[j*cols:(j + 1)*cols], mat[i*cols:(i + 1)*cols]

        def cross_cancel(a, i, b, j):
            """Does the row op row[i] = a*row[i] - b*row[j]"""
            q = (j - i)*cols
            for p in range(i*cols, (i + 1)*cols):
                mat[p] = a*mat[p] - b*mat[p + q]

        piv_row, piv_col = 0, 0
        pivot_cols = []
        swaps = []
        # use a fraction free method to zero above and below each pivot
        while piv_col < cols and piv_row < rows:
            pivot_offset, pivot_val, \
            assumed_nonzero, newly_determined = _find_reasonable_pivot(
                get_col(piv_col)[piv_row:], iszerofunc, simpfunc)

            # _find_reasonable_pivot may have simplified some things
            # in the process.  Let's not let them go to waste
            for (offset, val) in newly_determined:
                offset += piv_row
                mat[offset*cols + piv_col] = val

            if pivot_offset is None:
                piv_col += 1
                continue

            pivot_cols.append(piv_col)
            if pivot_offset != 0:
                row_swap(piv_row, pivot_offset + piv_row)
                swaps.append((piv_row, pivot_offset + piv_row))

            # if we aren't normalizing last, we normalize
            # before we zero the other rows
            if normalize_last is False:
                i, j = piv_row, piv_col
                mat[i*cols + j] = S.One
                for p in range(i*cols + j + 1, (i + 1)*cols):
                    mat[p] = mat[p] / pivot_val
                # after normalizing, the pivot value is 1
                pivot_val = S.One

            # zero above and below the pivot
            for row in range(rows):
                # don't zero our current row
                if row == piv_row:
                    continue
                # don't zero above the pivot unless we're told.
                if zero_above is False and row < piv_row:
                    continue
                # if we're already a zero, don't do anything
                val = mat[row*cols + piv_col]
                if iszerofunc(val):
                    continue

                cross_cancel(pivot_val, row, val, piv_row)
            piv_row += 1

        # normalize each row
        if normalize_last is True and normalize is True:
            for piv_i, piv_j in enumerate(pivot_cols):
                pivot_val = mat[piv_i*cols + piv_j]
                mat[piv_i*cols + piv_j] = S.One
                for p in range(piv_i*cols + piv_j + 1, (piv_i + 1)*cols):
                    mat[p] = mat[p] / pivot_val

        return self._new(self.rows, self.cols, mat), tuple(pivot_cols), tuple(swaps)

    def echelon_form(self, iszerofunc=_iszero, simplify=False, with_pivots=False):
        """Returns a matrix row-equivalent to `self` that is
        in echelon form.  Note that echelon form of a matrix
        is *not* unique, however, properties like the row
        space and the null space are preserved."""
        simpfunc = simplify if isinstance(
            simplify, FunctionType) else _simplify

        mat, pivots, swaps = self._eval_echelon_form(iszerofunc, simpfunc)
        if with_pivots:
            return mat, pivots
        return mat

    def elementary_col_op(self, op="n->kn", col=None, k=None, col1=None, col2=None):
        """Perfoms the elementary column operation `op`.

        `op` may be one of

            * "n->kn" (column n goes to k*n)
            * "n<->m" (swap column n and column m)
            * "n->n+km" (column n goes to column n + k*column m)

        Parameters
        =========

        op : string; the elementary row operation
        col : the column to apply the column operation
        k : the multiple to apply in the column operation
        col1 : one column of a column swap
        col2 : second column of a column swap or column "m" in the column operation
               "n->n+km"
        """

        op, col, k, col1, col2 = self._normalize_op_args(op, col, k, col1, col2, "col")

        # now that we've validated, we're all good to dispatch
        if op == "n->kn":
            return self._eval_col_op_multiply_col_by_const(col, k)
        if op == "n<->m":
            return self._eval_col_op_swap(col1, col2)
        if op == "n->n+km":
            return self._eval_col_op_add_multiple_to_other_col(col, k, col2)

    def elementary_row_op(self, op="n->kn", row=None, k=None, row1=None, row2=None):
        """Perfoms the elementary row operation `op`.

        `op` may be one of

            * "n->kn" (row n goes to k*n)
            * "n<->m" (swap row n and row m)
            * "n->n+km" (row n goes to row n + k*row m)

        Parameters
        ==========

        op : string; the elementary row operation
        row : the row to apply the row operation
        k : the multiple to apply in the row operation
        row1 : one row of a row swap
        row2 : second row of a row swap or row "m" in the row operation
               "n->n+km"
        """

        op, row, k, row1, row2 = self._normalize_op_args(op, row, k, row1, row2, "row")

        # now that we've validated, we're all good to dispatch
        if op == "n->kn":
            return self._eval_row_op_multiply_row_by_const(row, k)
        if op == "n<->m":
            return self._eval_row_op_swap(row1, row2)
        if op == "n->n+km":
            return self._eval_row_op_add_multiple_to_other_row(row, k, row2)

    @property
    def is_echelon(self, iszerofunc=_iszero):
        """Returns `True` if he matrix is in echelon form.
        That is, all rows of zeros are at the bottom, and below
        each leading non-zero in a row are exclusively zeros."""

        return self._eval_is_echelon(iszerofunc)

    def rank(self, iszerofunc=_iszero, simplify=False):
        """
        Returns the rank of a matrix

        >>> from sympy import Matrix
        >>> from sympy.abc import x
        >>> m = Matrix([[1, 2], [x, 1 - 1/x]])
        >>> m.rank()
        2
        >>> n = Matrix(3, 3, range(1, 10))
        >>> n.rank()
        2
        """
        simpfunc = simplify if isinstance(
            simplify, FunctionType) else _simplify

        # for small matrices, we compute the rank explicitly
        # if is_zero on elements doesn't answer the question
        # for small matrices, we fall back to the full routine.
        if self.rows <= 0 or self.cols <= 0:
            return 0
        if self.rows <= 1 or self.cols <= 1:
            zeros = [iszerofunc(x) for x in self]
            if False in zeros:
                return 1
        if self.rows == 2 and self.cols == 2:
            zeros = [iszerofunc(x) for x in self]
            if not False in zeros and not None in zeros:
                return 0
            det = self.det()
            if iszerofunc(det) and False in zeros:
                return 1
            if iszerofunc(det) is False:
                return 2

        mat, _ = self._permute_complexity_right(iszerofunc=iszerofunc)
        echelon_form, pivots, swaps = mat._eval_echelon_form(iszerofunc=iszerofunc, simpfunc=simpfunc)
        return len(pivots)

    def rref(self, iszerofunc=_iszero, simplify=False, pivots=True, normalize_last=True):
        """Return reduced row-echelon form of matrix and indices of pivot vars.

        Parameters
        ==========

        iszerofunc : Function
            A function used for detecting whether an element can
            act as a pivot.  `lambda x: x.is_zero` is used by default.
        simplify : Function
            A function used to simplify elements when looking for a pivot.
            By default SymPy's `simplify`is used.
        pivots : True or False
            If `True`, a tuple containing the row-reduced matrix and a tuple
            of pivot columns is returned.  If `False` just the row-reduced
            matrix is returned.
        normalize_last : True or False
            If `True`, no pivots are normalized to `1` until after all entries
            above and below each pivot are zeroed.  This means the row
            reduction algorithm is fraction free until the very last step.
            If `False`, the naive row reduction procedure is used where
            each pivot is normalized to be `1` before row operations are
            used to zero above and below the pivot.

        Notes
        =====

        The default value of `normalize_last=True` can provide significant
        speedup to row reduction, especially on matrices with symbols.  However,
        if you depend on the form row reduction algorithm leaves entries
        of the matrix, set `noramlize_last=False`


        Examples
        ========

        >>> from sympy import Matrix
        >>> from sympy.abc import x
        >>> m = Matrix([[1, 2], [x, 1 - 1/x]])
        >>> m.rref()
        (Matrix([
        [1, 0],
        [0, 1]]), (0, 1))
        >>> rref_matrix, rref_pivots = m.rref()
        >>> rref_matrix
        Matrix([
        [1, 0],
        [0, 1]])
        >>> rref_pivots
        (0, 1)
        """
        simpfunc = simplify if isinstance(
            simplify, FunctionType) else _simplify

        ret, pivot_cols = self._eval_rref(iszerofunc=iszerofunc,
                                          simpfunc=simpfunc,
                                          normalize_last=normalize_last)
        if pivots:
            ret = (ret, pivot_cols)
        return ret


class MatrixSubspaces(MatrixReductions):
    """Provides methods relating to the fundamental subspaces
    of a matrix.  Should not be instantiated directly."""

    def columnspace(self, simplify=False):
        """Returns a list of vectors (Matrix objects) that span columnspace of self

        Examples
        ========

        >>> from sympy.matrices import Matrix
        >>> m = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6])
        >>> m
        Matrix([
        [ 1,  3, 0],
        [-2, -6, 0],
        [ 3,  9, 6]])
        >>> m.columnspace()
        [Matrix([
        [ 1],
        [-2],
        [ 3]]), Matrix([
        [0],
        [0],
        [6]])]

        See Also
        ========

        nullspace
        rowspace
        """
        reduced, pivots = self.echelon_form(simplify=simplify, with_pivots=True)

        return [self.col(i) for i in pivots]

    def nullspace(self, simplify=False):
        """Returns list of vectors (Matrix objects) that span nullspace of self

        Examples
        ========

        >>> from sympy.matrices import Matrix
        >>> m = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6])
        >>> m
        Matrix([
        [ 1,  3, 0],
        [-2, -6, 0],
        [ 3,  9, 6]])
        >>> m.nullspace()
        [Matrix([
        [-3],
        [ 1],
        [ 0]])]

        See Also
        ========

        columnspace
        rowspace
        """

        reduced, pivots = self.rref(simplify=simplify)

        free_vars = [i for i in range(self.cols) if i not in pivots]

        basis = []
        for free_var in free_vars:
            # for each free variable, we will set it to 1 and all others
            # to 0.  Then, we will use back substitution to solve the system
            vec = [S.Zero]*self.cols
            vec[free_var] = S.One
            for piv_row, piv_col in enumerate(pivots):
                for pos in pivots[piv_row+1:] + (free_var,):
                    vec[piv_col] -= reduced[piv_row, pos]
            basis.append(vec)

        return [self._new(self.cols, 1, b) for b in basis]

    def rowspace(self, simplify=False):
        """Returns a list of vectors that span the row space of self."""

        reduced, pivots = self.echelon_form(simplify=simplify, with_pivots=True)

        return [reduced.row(i) for i in range(len(pivots))]

    @classmethod
    def orthogonalize(cls, *vecs, **kwargs):
        """Apply the Gram-Schmidt orthogonalization procedure
        to vectors supplied in `vecs`.

        Arguments
        =========

        vecs : vectors to be made orthogonal
        normalize : bool. Whether the returned vectors
                    should be renormalized to be unit vectors.
        """

        normalize = kwargs.get('normalize', False)

        def project(a, b):
            return b * (a.dot(b) / b.dot(b))

        def perp_to_subspace(vec, basis):
            """projects vec onto the subspace given
            by the orthogonal basis `basis`"""
            components = [project(vec, b) for b in basis]
            if len(basis) == 0:
                return vec
            return vec - reduce(lambda a, b: a + b, components)

        ret = []
        # make sure we start with a non-zero vector
        while len(vecs) > 0 and vecs[0].is_zero:
            del vecs[0]

        for vec in vecs:
            perp = perp_to_subspace(vec, ret)
            if not perp.is_zero:
                ret.append(perp)

        if normalize:
            ret = [vec / vec.norm() for vec in ret]

        return ret


class MatrixEigen(MatrixSubspaces):
    """Provides basic matrix eigenvalue/vector operations.
    Should not be instantiated directly."""

    _cache_is_diagonalizable = None
    _cache_eigenvects = None

    def diagonalize(self, reals_only=False, sort=False, normalize=False):
        """
        Return (P, D), where D is diagonal and

            D = P^-1 * M * P

        where M is current matrix.

        Parameters
        ==========

        reals_only : bool. Whether to throw an error if complex numbers are need
                     to diagonalize. (Default: False)
        sort : bool. Sort the eigenvalues along the diagonal. (Default: False)
        normalize : bool. If True, normalize the columns of P. (Default: False)

        Examples
        ========

        >>> from sympy import Matrix
        >>> m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
        >>> m
        Matrix([
        [1,  2, 0],
        [0,  3, 0],
        [2, -4, 2]])
        >>> (P, D) = m.diagonalize()
        >>> D
        Matrix([
        [1, 0, 0],
        [0, 2, 0],
        [0, 0, 3]])
        >>> P
        Matrix([
        [-1, 0, -1],
        [ 0, 0, -1],
        [ 2, 1,  2]])
        >>> P.inv() * m * P
        Matrix([
        [1, 0, 0],
        [0, 2, 0],
        [0, 0, 3]])

        See Also
        ========

        is_diagonal
        is_diagonalizable
        """

        if not self.is_square:
            raise NonSquareMatrixError()

        if not self.is_diagonalizable(reals_only=reals_only, clear_cache=False):
            raise MatrixError("Matrix is not diagonalizable")

        eigenvecs = self._cache_eigenvects
        if eigenvecs is None:
            eigenvecs = self.eigenvects(simplify=True)

        if sort:
            eigenvecs = sorted(eigenvecs, key=default_sort_key)

        p_cols, diag = [], []
        for val, mult, basis in eigenvecs:
            diag += [val] * mult
            p_cols += basis

        if normalize:
            p_cols = [v / v.norm() for v in p_cols]

        return self.hstack(*p_cols), self.diag(*diag)

    def eigenvals(self, error_when_incomplete=True, **flags):
        """Return eigenvalues using the Berkowitz agorithm to compute
        the characteristic polynomial.

        Parameters
        ==========

        error_when_incomplete : bool
            Raise an error when not all eigenvalues are computed. This is
            caused by ``roots`` not returning a full list of eigenvalues.

        Since the roots routine doesn't always work well with Floats,
        they will be replaced with Rationals before calling that
        routine. If this is not desired, set flag ``rational`` to False.
        """
        mat = self
        if not mat:
            return {}
        if flags.pop('rational', True):
            if any(v.has(Float) for v in mat):
                mat = mat.applyfunc(lambda x: nsimplify(x, rational=True))

        flags.pop('simplify', None)  # pop unsupported flag
        eigs = roots(mat.charpoly(x=Dummy('x')), **flags)

        # make sure the algebraic multiplicty sums to the
        # size of the matrix
        if error_when_incomplete and sum(m for m in eigs.values()) != self.cols:
            raise MatrixError("Could not compute eigenvalues for {}".format(self))

        return eigs

    def eigenvects(self, error_when_incomplete=True, **flags):
        """Return list of triples (eigenval, multiplicity, basis).

        The flag ``simplify`` has two effects:
            1) if bool(simplify) is True, as_content_primitive()
            will be used to tidy up normalization artifacts;
            2) if nullspace needs simplification to compute the
            basis, the simplify flag will be passed on to the
            nullspace routine which will interpret it there.

        Parameters
        ==========

        error_when_incomplete : bool
            Raise an error when not all eigenvalues are computed. This is
            caused by ``roots`` not returning a full list of eigenvalues.

        If the matrix contains any Floats, they will be changed to Rationals
        for computation purposes, but the answers will be returned after being
        evaluated with evalf. If it is desired to removed small imaginary
        portions during the evalf step, pass a value for the ``chop`` flag.
        """
        from sympy.matrices import eye

        simplify = flags.get('simplify', True)
        if not isinstance(simplify, FunctionType):
            simpfunc = _simplify if simplify else lambda x: x
        primitive = flags.get('simplify', False)
        chop = flags.pop('chop', False)

        flags.pop('multiple', None)  # remove this if it's there

        mat = self
        # roots doesn't like Floats, so replace them with Rationals
        has_floats = any(v.has(Float) for v in self)
        if has_floats:
            mat = mat.applyfunc(lambda x: nsimplify(x, rational=True))

        def eigenspace(eigenval):
            """Get a basis for the eigenspace for a particular eigenvalue"""
            m = mat - self.eye(mat.rows) * eigenval
            ret = m.nullspace()
            # the nullspace for a real eigenvalue should be
            # non-trivial.  If we didn't find an eigenvector, try once
            # more a little harder
            if len(ret) == 0 and simplify:
                ret = m.nullspace(simplify=True)
            if len(ret) == 0:
                raise NotImplementedError(
                        "Can't evaluate eigenvector for eigenvalue %s" % eigenval)
            return ret

        eigenvals = mat.eigenvals(rational=False,
                                  error_when_incomplete=error_when_incomplete,
                                  **flags)
        ret = [(val, mult, eigenspace(val)) for val, mult in
                    sorted(eigenvals.items(), key=default_sort_key)]
        if primitive:
            # if the primitive flag is set, get rid of any common
            # integer denominators
            def denom_clean(l):
                from sympy import gcd
                return [(v / gcd(list(v))).applyfunc(simpfunc) for v in l]
            ret = [(val, mult, denom_clean(es)) for val, mult, es in ret]
        if has_floats:
            # if we had floats to start with, turn the eigenvectors to floats
            ret = [(val.evalf(chop=chop), mult, [v.evalf(chop=chop) for v in es]) for val, mult, es in ret]
        return ret

    def is_diagonalizable(self, reals_only=False, **kwargs):
        """Returns true if a matrix is diagonalizable.

        Parameters
        ==========

        reals_only : bool. If reals_only=True, determine whether the matrix can be
                     diagonalized without complex numbers. (Default: False)

        kwargs
        ======

        clear_cache : bool. If True, clear the result of any computations when finished.
                      (Default: True)

        Examples
        ========

        >>> from sympy import Matrix
        >>> m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
        >>> m
        Matrix([
        [1,  2, 0],
        [0,  3, 0],
        [2, -4, 2]])
        >>> m.is_diagonalizable()
        True
        >>> m = Matrix(2, 2, [0, 1, 0, 0])
        >>> m
        Matrix([
        [0, 1],
        [0, 0]])
        >>> m.is_diagonalizable()
        False
        >>> m = Matrix(2, 2, [0, 1, -1, 0])
        >>> m
        Matrix([
        [ 0, 1],
        [-1, 0]])
        >>> m.is_diagonalizable()
        True
        >>> m.is_diagonalizable(reals_only=True)
        False

        See Also
        ========

        is_diagonal
        diagonalize
        """

        clear_cache = kwargs.get('clear_cache', True)
        if 'clear_subproducts' in kwargs:
            clear_cache = kwargs.get('clear_subproducts')

        def cleanup():
            """Clears any cached values if requested"""
            if clear_cache:
                self._cache_eigenvects = None
                self._cache_is_diagonalizable = None

        if not self.is_square:
            cleanup()
            return False

        # use the cached value if we have it
        if self._cache_is_diagonalizable is not None:
            ret = self._cache_is_diagonalizable
            cleanup()
            return ret

        if all(e.is_real for e in self) and self.is_symmetric():
            # every real symmetric matrix is real diagonalizable
            self._cache_is_diagonalizable = True
            cleanup()
            return True

        self._cache_eigenvects = self.eigenvects(simplify=True)
        ret = True
        for val, mult, basis in self._cache_eigenvects:
            # if we have a complex eigenvalue
            if reals_only and not val.is_real:
                ret = False
            # if the geometric multiplicity doesn't equal the algebraic
            if mult != len(basis):
                ret = False
        cleanup()
        return ret

    def jordan_form(self, calc_transform=True, **kwargs):
        """Return `(P, J)` where `J` is a Jordan block
        matrix and `P` is a matrix such that

            `self == P*J*P**-1`


        Parameters
        ==========

        calc_transform : bool
            If ``False``, then only `J` is returned.
        chop : bool
            All matrices are convered to exact types when computing
            eigenvalues and eigenvectors.  As a result, there may be
            approximation errors.  If ``chop==True``, these errors
            will be truncated.

        Examples
        ========

        >>> from sympy import Matrix
        >>> m = Matrix([[ 6,  5, -2, -3], [-3, -1,  3,  3], [ 2,  1, -2, -3], [-1,  1,  5,  5]])
        >>> P, J = m.jordan_form()
        >>> J
        Matrix([
        [2, 1, 0, 0],
        [0, 2, 0, 0],
        [0, 0, 2, 1],
        [0, 0, 0, 2]])

        See Also
        ========

        jordan_block
        """
        if not self.is_square:
            raise NonSquareMatrixError("Only square matrices have Jordan forms")

        chop = kwargs.pop('chop', False)
        mat = self
        has_floats = any(v.has(Float) for v in self)

        def restore_floats(*args):
            """If `has_floats` is `True`, cast all `args` as
            matrices of floats."""
            if has_floats:
                args = [m.evalf(chop=chop) for m in args]
            if len(args) == 1:
                return args[0]
            return args

        # cache calculations for some speedup
        mat_cache = {}
        def eig_mat(val, pow):
            """Cache computations of (self - val*I)**pow for quick
            retrieval"""
            if (val, pow) in mat_cache:
                return mat_cache[(val, pow)]
            if (val, pow - 1) in mat_cache:
                mat_cache[(val, pow)] = mat_cache[(val, pow - 1)] * mat_cache[(val, 1)]
            else:
                mat_cache[(val, pow)] = (mat - val*self.eye(self.rows))**pow
            return mat_cache[(val, pow)]

        # helper functions
        def nullity_chain(val):
            """Calculate the sequence  [0, nullity(E), nullity(E**2), ...]
            until it is constant where `E = self - val*I`"""
            # mat.rank() is faster than computing the null space,
            # so use the rank-nullity theorem
            cols = self.cols
            ret = [0]
            nullity = cols - eig_mat(val, 1).rank()
            i = 2
            while nullity != ret[-1]:
                ret.append(nullity)
                nullity = cols - eig_mat(val, i).rank()
                i += 1
            return ret

        def blocks_from_nullity_chain(d):
            """Return a list of the size of each Jordan block.
            If d_n is the nullity of E**n, then the number
            of Jordan blocks of size n is

                2*d_n - d_(n-1) - d_(n+1)"""
            # d[0] is always the number of columns, so skip past it
            mid = [2*d[n] - d[n - 1] - d[n + 1] for n in range(1, len(d) - 1)]
            # d is assumed to plateau with "d[ len(d) ] == d[-1]", so
            # 2*d_n - d_(n-1) - d_(n+1) == d_n - d_(n-1)
            end = [d[-1] - d[-2]] if len(d) > 1 else [d[0]]
            return mid + end

        def pick_vec(small_basis, big_basis):
            """Picks a vector from big_basis that isn't in
            the subspace spanned by small_basis"""
            if len(small_basis) == 0:
                return big_basis[0]
            for v in big_basis:
                _, pivots = self.hstack(*(small_basis + [v])).echelon_form(with_pivots=True)
                if pivots[-1] == len(small_basis):
                    return v

        # roots doesn't like Floats, so replace them with Rationals
        if has_floats:
            mat = mat.applyfunc(lambda x: nsimplify(x, rational=True))

        # first calculate the jordan block structure
        eigs = mat.eigenvals()

        # make sure that we found all the roots by counting
        # the algebraic multiplicity
        if sum(m for m in eigs.values()) != mat.cols:
            raise MatrixError("Could not compute eigenvalues for {}".format(mat))

        # most matrices have distinct eigenvalues
        # and so are diagonalizable.  In this case, don't
        # do extra work!
        if len(eigs.keys()) == mat.cols:
            blocks = list(sorted(eigs.keys(), key=default_sort_key))
            jordan_mat = mat.diag(*blocks)
            if not calc_transform:
                return restore_floats(jordan_mat)
            jordan_basis = [eig_mat(eig, 1).nullspace()[0] for eig in blocks]
            basis_mat = mat.hstack(*jordan_basis)
            return restore_floats(basis_mat, jordan_mat)

        block_structure = []
        for eig in sorted(eigs.keys(), key=default_sort_key):
            chain = nullity_chain(eig)
            block_sizes = blocks_from_nullity_chain(chain)
            # if block_sizes == [a, b, c, ...], then the number of
            # Jordan blocks of size 1 is a, of size 2 is b, etc.
            # create an array that has (eig, block_size) with one
            # entry for each block
            size_nums = [(i+1, num) for i, num in enumerate(block_sizes)]
            # we expect larger Jordan blocks to come earlier
            size_nums.reverse()

            block_structure.extend(
                (eig, size) for size, num in size_nums for _ in range(num))
        blocks = (mat.jordan_block(size=size, eigenvalue=eig) for eig, size in block_structure)
        jordan_mat = mat.diag(*blocks)

        if not calc_transform:
            return restore_floats(jordan_mat)

        # For each generalized eigenspace, calculate a basis.
        # We start by looking for a vector in null( (A - eig*I)**n )
        # which isn't in null( (A - eig*I)**(n-1) ) where n is
        # the size of the Jordan block
        #
        # Ideally we'd just loop through block_structure and
        # compute each generalized eigenspace.  However, this
        # causes a lot of unneeded computation.  Instead, we
        # go through the eigenvalues separately, since we know
        # their generalized eigenspaces must have bases that
        # are linearly independent.
        jordan_basis = []

        for eig in sorted(eigs.keys(), key=default_sort_key):
            eig_basis = []
            for block_eig, size in block_structure:
                if block_eig != eig:
                    continue
                null_big = (eig_mat(eig, size)).nullspace()
                null_small = (eig_mat(eig, size - 1)).nullspace()
                # we want to pick something that is in the big basis
                # and not the small, but also something that is independent
                # of any other generalized eigenvectors from a different
                # generalized eigenspace sharing the same eigenvalue.
                vec = pick_vec(null_small + eig_basis, null_big)
                new_vecs = [(eig_mat(eig, i))*vec for i in range(size)]
                eig_basis.extend(new_vecs)
                jordan_basis.extend(reversed(new_vecs))

        basis_mat = mat.hstack(*jordan_basis)

        return restore_floats(basis_mat, jordan_mat)

    def left_eigenvects(self, **flags):
        """Returns left eigenvectors and eigenvalues.

        This function returns the list of triples (eigenval, multiplicity,
        basis) for the left eigenvectors. Options are the same as for
        eigenvects(), i.e. the ``**flags`` arguments gets passed directly to
        eigenvects().

        Examples
        ========

        >>> from sympy import Matrix
        >>> M = Matrix([[0, 1, 1], [1, 0, 0], [1, 1, 1]])
        >>> M.eigenvects()
        [(-1, 1, [Matrix([
        [-1],
        [ 1],
        [ 0]])]), (0, 1, [Matrix([
        [ 0],
        [-1],
        [ 1]])]), (2, 1, [Matrix([
        [2/3],
        [1/3],
        [  1]])])]
        >>> M.left_eigenvects()
        [(-1, 1, [Matrix([[-2, 1, 1]])]), (0, 1, [Matrix([[-1, -1, 1]])]), (2,
        1, [Matrix([[1, 1, 1]])])]

        """
        eigs = self.transpose().eigenvects(**flags)

        return [(val, mult, [l.transpose() for l in basis]) for val, mult, basis in eigs]

    def singular_values(self):
        """Compute the singular values of a Matrix

        Examples
        ========

        >>> from sympy import Matrix, Symbol
        >>> x = Symbol('x', real=True)
        >>> A = Matrix([[0, 1, 0], [0, x, 0], [-1, 0, 0]])
        >>> A.singular_values()
        [sqrt(x**2 + 1), 1, 0]

        See Also
        ========

        condition_number
        """
        mat = self
        # Compute eigenvalues of A.H A
        valmultpairs = (mat.H * mat).eigenvals()

        # Expands result from eigenvals into a simple list
        vals = []
        for k, v in valmultpairs.items():
            vals += [sqrt(k)] * v  # dangerous! same k in several spots!
        # sort them in descending order
        vals.sort(reverse=True, key=default_sort_key)

        return vals



class MatrixCalculus(MatrixCommon):
    """Provides calculus-related matrix operations."""

    def diff(self, *args):
        """Calculate the derivative of each element in the matrix.
        ``args`` will be passed to the ``integrate`` function.

        Examples
        ========

        >>> from sympy.matrices import Matrix
        >>> from sympy.abc import x, y
        >>> M = Matrix([[x, y], [1, 0]])
        >>> M.diff(x)
        Matrix([
        [1, 0],
        [0, 0]])

        See Also
        ========

        integrate
        limit
        """
        return self.applyfunc(lambda x: x.diff(*args))

    def integrate(self, *args):
        """Integrate each element of the matrix.  ``args`` will
        be passed to the ``integrate`` function.

        Examples
        ========

        >>> from sympy.matrices import Matrix
        >>> from sympy.abc import x, y
        >>> M = Matrix([[x, y], [1, 0]])
        >>> M.integrate((x, ))
        Matrix([
        [x**2/2, x*y],
        [     x,   0]])
        >>> M.integrate((x, 0, 2))
        Matrix([
        [2, 2*y],
        [2,   0]])

        See Also
        ========

        limit
        diff
        """
        return self.applyfunc(lambda x: x.integrate(*args))

    def jacobian(self, X):
        """Calculates the Jacobian matrix (derivative of a vector-valued function).

        Parameters
        ==========

        self : vector of expressions representing functions f_i(x_1, ..., x_n).
        X : set of x_i's in order, it can be a list or a Matrix

        Both self and X can be a row or a column matrix in any order
        (i.e., jacobian() should always work).

        Examples
        ========

        >>> from sympy import sin, cos, Matrix
        >>> from sympy.abc import rho, phi
        >>> X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
        >>> Y = Matrix([rho, phi])
        >>> X.jacobian(Y)
        Matrix([
        [cos(phi), -rho*sin(phi)],
        [sin(phi),  rho*cos(phi)],
        [   2*rho,             0]])
        >>> X = Matrix([rho*cos(phi), rho*sin(phi)])
        >>> X.jacobian(Y)
        Matrix([
        [cos(phi), -rho*sin(phi)],
        [sin(phi),  rho*cos(phi)]])

        See Also
        ========

        hessian
        wronskian
        """
        if not isinstance(X, MatrixBase):
            X = self._new(X)
        # Both X and self can be a row or a column matrix, so we need to make
        # sure all valid combinations work, but everything else fails:
        if self.shape[0] == 1:
            m = self.shape[1]
        elif self.shape[1] == 1:
            m = self.shape[0]
        else:
            raise TypeError("self must be a row or a column matrix")
        if X.shape[0] == 1:
            n = X.shape[1]
        elif X.shape[1] == 1:
            n = X.shape[0]
        else:
            raise TypeError("X must be a row or a column matrix")

        # m is the number of functions and n is the number of variables
        # computing the Jacobian is now easy:
        return self._new(m, n, lambda j, i: self[j].diff(X[i]))

    def limit(self, *args):
        """Calculate the limit of each element in the matrix.
        ``args`` will be passed to the ``limit`` function.

        Examples
        ========

        >>> from sympy.matrices import Matrix
        >>> from sympy.abc import x, y
        >>> M = Matrix([[x, y], [1, 0]])
        >>> M.limit(x, 2)
        Matrix([
        [2, y],
        [1, 0]])

        See Also
        ========

        integrate
        diff
        """
        return self.applyfunc(lambda x: x.limit(*args))


# https://github.com/sympy/sympy/pull/12854
class MatrixDeprecated(MatrixCommon):
    """A class to house deprecated matrix methods."""

    def berkowitz_charpoly(self, x=Dummy('lambda'), simplify=_simplify):
        return self.charpoly(x=x)

    def berkowitz_det(self):
        """Computes determinant using Berkowitz method.

        See Also
        ========

        det
        berkowitz
        """
        return self.det(method='berkowitz')

    def berkowitz_eigenvals(self, **flags):
        """Computes eigenvalues of a Matrix using Berkowitz method.

        See Also
        ========

        berkowitz
        """
        return self.eigenvals(**flags)

    def berkowitz_minors(self):
        """Computes principal minors using Berkowitz method.

        See Also
        ========

        berkowitz
        """
        sign, minors = S.One, []

        for poly in self.berkowitz():
            minors.append(sign * poly[-1])
            sign = -sign

        return tuple(minors)

    def berkowitz(self):
        from sympy.matrices import zeros
        berk = ((1,),)
        if not self:
            return berk

        if not self.is_square:
            raise NonSquareMatrixError()

        A, N = self, self.rows
        transforms = [0] * (N - 1)

        for n in range(N, 1, -1):
            T, k = zeros(n + 1, n), n - 1

            R, C = -A[k, :k], A[:k, k]
            A, a = A[:k, :k], -A[k, k]

            items = [C]

            for i in range(0, n - 2):
                items.append(A * items[i])

            for i, B in enumerate(items):
                items[i] = (R * B)[0, 0]

            items = [S.One, a] + items

            for i in range(n):
                T[i:, i] = items[:n - i + 1]

            transforms[k - 1] = T

        polys = [self._new([S.One, -A[0, 0]])]

        for i, T in enumerate(transforms):
            polys.append(T * polys[i])

        return berk + tuple(map(tuple, polys))

    def cofactorMatrix(self, method="berkowitz"):
        return self.cofactor_matrix(method=method)

    def det_bareis(self):
        return self.det(method='bareiss')

    def det_bareiss(self):
        """Compute matrix determinant using Bareiss' fraction-free
        algorithm which is an extension of the well known Gaussian
        elimination method. This approach is best suited for dense
        symbolic matrices and will result in a determinant with
        minimal number of fractions. It means that less term
        rewriting is needed on resulting formulae.

        TODO: Implement algorithm for sparse matrices (SFF),
        http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.

        See Also
        ========

        det
        berkowitz_det
        """
        return self.det(method='bareiss')

    def det_LU_decomposition(self):
        """Compute matrix determinant using LU decomposition


        Note that this method fails if the LU decomposition itself
        fails. In particular, if the matrix has no inverse this method
        will fail.

        TODO: Implement algorithm for sparse matrices (SFF),
        http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.

        See Also
        ========


        det
        det_bareiss
        berkowitz_det
        """
        return self.det(method='lu')

    def jordan_cell(self, eigenval, n):
        return self.jordan_block(size=n, eigenvalue=eigenval)

    def jordan_cells(self, calc_transformation=True):
        P, J = self.jordan_form()
        return P, J.get_diag_blocks()

    def minorEntry(self, i, j, method="berkowitz"):
        return self.minor(i, j, method=method)

    def minorMatrix(self, i, j):
        return self.minor_submatrix(i, j)

    def permuteBkwd(self, perm):
        """Permute the rows of the matrix with the given permutation in reverse."""
        return self.permute_rows(perm, direction='backward')

    def permuteFwd(self, perm):
        """Permute the rows of the matrix with the given permutation."""
        return self.permute_rows(perm, direction='forward')


[docs]class MatrixBase(MatrixDeprecated, MatrixCalculus, MatrixEigen, MatrixCommon): """Base class for matrix objects.""" # Added just for numpy compatibility __array_priority__ = 11 is_Matrix = True _class_priority = 3 _sympify = staticmethod(sympify) __hash__ = None # Mutable def __array__(self): from .dense import matrix2numpy return matrix2numpy(self) def __getattr__(self, attr): if attr in ('diff', 'integrate', 'limit'): def doit(*args): item_doit = lambda item: getattr(item, attr)(*args) return self.applyfunc(item_doit) return doit else: raise AttributeError( "%s has no attribute %s." % (self.__class__.__name__, attr)) def __len__(self): """Return the number of elements of self. Implemented mainly so bool(Matrix()) == False. """ return self.rows * self.cols def __mathml__(self): mml = "" for i in range(self.rows): mml += "<matrixrow>" for j in range(self.cols): mml += self[i, j].__mathml__() mml += "</matrixrow>" return "<matrix>" + mml + "</matrix>" # needed for python 2 compatibility def __ne__(self, other): return not self == other def _matrix_pow_by_jordan_blocks(self, num): from sympy.matrices import diag, MutableMatrix from sympy import binomial def jordan_cell_power(jc, n): N = jc.shape[0] l = jc[0, 0] if l == 0 and (n < N - 1) != False: raise ValueError("Matrix det == 0; not invertible") elif l == 0 and N > 1 and n % 1 != 0: raise ValueError("Non-integer power cannot be evaluated") for i in range(N): for j in range(N-i): bn = binomial(n, i) if isinstance(bn, binomial): bn = bn._eval_expand_func() jc[j, i+j] = l**(n-i)*bn P, J = self.jordan_form() jordan_cells = J.get_diag_blocks() # Make sure jordan_cells matrices are mutable: jordan_cells = [MutableMatrix(j) for j in jordan_cells] for j in jordan_cells: jordan_cell_power(j, num) return self._new(P*diag(*jordan_cells)*P.inv()) def __repr__(self): return sstr(self) def __str__(self): if self.rows == 0 or self.cols == 0: return 'Matrix(%s, %s, [])' % (self.rows, self.cols) return "Matrix(%s)" % str(self.tolist()) def _diagonalize_clear_subproducts(self): del self._is_symbolic del self._is_symmetric del self._eigenvects def _format_str(self, printer=None): if not printer: from sympy.printing.str import StrPrinter printer = StrPrinter() # Handle zero dimensions: if self.rows == 0 or self.cols == 0: return 'Matrix(%s, %s, [])' % (self.rows, self.cols) if self.rows == 1: return "Matrix([%s])" % self.table(printer, rowsep=',\n') return "Matrix([\n%s])" % self.table(printer, rowsep=',\n') @classmethod def _handle_creation_inputs(cls, *args, **kwargs): """Return the number of rows, cols and flat matrix elements. Examples ======== >>> from sympy import Matrix, I Matrix can be constructed as follows: * from a nested list of iterables >>> Matrix( ((1, 2+I), (3, 4)) ) Matrix([ [1, 2 + I], [3, 4]]) * from un-nested iterable (interpreted as a column) >>> Matrix( [1, 2] ) Matrix([ [1], [2]]) * from un-nested iterable with dimensions >>> Matrix(1, 2, [1, 2] ) Matrix([[1, 2]]) * from no arguments (a 0 x 0 matrix) >>> Matrix() Matrix(0, 0, []) * from a rule >>> Matrix(2, 2, lambda i, j: i/(j + 1) ) Matrix([ [0, 0], [1, 1/2]]) """ from sympy.matrices.sparse import SparseMatrix flat_list = None if len(args) == 1: # Matrix(SparseMatrix(...)) if isinstance(args[0], SparseMatrix): return args[0].rows, args[0].cols, flatten(args[0].tolist()) # Matrix(Matrix(...)) elif isinstance(args[0], MatrixBase): return args[0].rows, args[0].cols, args[0]._mat # Matrix(MatrixSymbol('X', 2, 2)) elif isinstance(args[0], Basic) and args[0].is_Matrix: return args[0].rows, args[0].cols, args[0].as_explicit()._mat # Matrix(numpy.ones((2, 2))) elif hasattr(args[0], "__array__"): # NumPy array or matrix or some other object that implements # __array__. So let's first use this method to get a # numpy.array() and then make a python list out of it. arr = args[0].__array__() if len(arr.shape) == 2: rows, cols = arr.shape[0], arr.shape[1] flat_list = [cls._sympify(i) for i in arr.ravel()] return rows, cols, flat_list elif len(arr.shape) == 1: rows, cols = arr.shape[0], 1 flat_list = [S.Zero] * rows for i in range(len(arr)): flat_list[i] = cls._sympify(arr[i]) return rows, cols, flat_list else: raise NotImplementedError( "SymPy supports just 1D and 2D matrices") # Matrix([1, 2, 3]) or Matrix([[1, 2], [3, 4]]) elif is_sequence(args[0]) \ and not isinstance(args[0], DeferredVector): in_mat = [] ncol = set() for row in args[0]: if isinstance(row, MatrixBase): in_mat.extend(row.tolist()) if row.cols or row.rows: # only pay attention if it's not 0x0 ncol.add(row.cols) else: in_mat.append(row) try: ncol.add(len(row)) except TypeError: ncol.add(1) if len(ncol) > 1: raise ValueError("Got rows of variable lengths: %s" % sorted(list(ncol))) cols = ncol.pop() if ncol else 0 rows = len(in_mat) if cols else 0 if rows: if not is_sequence(in_mat[0]): cols = 1 flat_list = [cls._sympify(i) for i in in_mat] return rows, cols, flat_list flat_list = [] for j in range(rows): for i in range(cols): flat_list.append(cls._sympify(in_mat[j][i])) elif len(args) == 3: rows = as_int(args[0]) cols = as_int(args[1]) if rows < 0 or cols < 0: raise ValueError("Cannot create a {} x {} matrix. " "Both dimensions must be positive".format(rows, cols)) # Matrix(2, 2, lambda i, j: i+j) if len(args) == 3 and isinstance(args[2], collections.Callable): op = args[2] flat_list = [] for i in range(rows): flat_list.extend( [cls._sympify(op(cls._sympify(i), cls._sympify(j))) for j in range(cols)]) # Matrix(2, 2, [1, 2, 3, 4]) elif len(args) == 3 and is_sequence(args[2]): flat_list = args[2] if len(flat_list) != rows * cols: raise ValueError( 'List length should be equal to rows*columns') flat_list = [cls._sympify(i) for i in flat_list] # Matrix() elif len(args) == 0: # Empty Matrix rows = cols = 0 flat_list = [] if flat_list is None: raise TypeError("Data type not understood") return rows, cols, flat_list def _setitem(self, key, value): """Helper to set value at location given by key. Examples ======== >>> from sympy import Matrix, I, zeros, ones >>> m = Matrix(((1, 2+I), (3, 4))) >>> m Matrix([ [1, 2 + I], [3, 4]]) >>> m[1, 0] = 9 >>> m Matrix([ [1, 2 + I], [9, 4]]) >>> m[1, 0] = [[0, 1]] To replace row r you assign to position r*m where m is the number of columns: >>> M = zeros(4) >>> m = M.cols >>> M[3*m] = ones(1, m)*2; M Matrix([ [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, 2, 2, 2]]) And to replace column c you can assign to position c: >>> M[2] = ones(m, 1)*4; M Matrix([ [0, 0, 4, 0], [0, 0, 4, 0], [0, 0, 4, 0], [2, 2, 4, 2]]) """ from .dense import Matrix is_slice = isinstance(key, slice) i, j = key = self.key2ij(key) is_mat = isinstance(value, MatrixBase) if type(i) is slice or type(j) is slice: if is_mat: self.copyin_matrix(key, value) return if not isinstance(value, Expr) and is_sequence(value): self.copyin_list(key, value) return raise ValueError('unexpected value: %s' % value) else: if (not is_mat and not isinstance(value, Basic) and is_sequence(value)): value = Matrix(value) is_mat = True if is_mat: if is_slice: key = (slice(*divmod(i, self.cols)), slice(*divmod(j, self.cols))) else: key = (slice(i, i + value.rows), slice(j, j + value.cols)) self.copyin_matrix(key, value) else: return i, j, self._sympify(value) return
[docs] def add(self, b): """Return self + b """ return self + b
[docs] def cholesky_solve(self, rhs): """Solves Ax = B using Cholesky decomposition, for a general square non-singular matrix. For a non-square matrix with rows > cols, the least squares solution is returned. See Also ======== lower_triangular_solve upper_triangular_solve gauss_jordan_solve diagonal_solve LDLsolve LUsolve QRsolve pinv_solve """ if self.is_symmetric(): L = self._cholesky() elif self.rows >= self.cols: L = (self.T * self)._cholesky() rhs = self.T * rhs else: raise NotImplementedError('Under-determined System. ' 'Try M.gauss_jordan_solve(rhs)') Y = L._lower_triangular_solve(rhs) return (L.T)._upper_triangular_solve(Y)
[docs] def cholesky(self): """Returns the Cholesky decomposition L of a matrix A such that L * L.T = A A must be a square, symmetric, positive-definite and non-singular matrix. Examples ======== >>> from sympy.matrices import Matrix >>> A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11))) >>> A.cholesky() Matrix([ [ 5, 0, 0], [ 3, 3, 0], [-1, 1, 3]]) >>> A.cholesky() * A.cholesky().T Matrix([ [25, 15, -5], [15, 18, 0], [-5, 0, 11]]) See Also ======== LDLdecomposition LUdecomposition QRdecomposition """ if not self.is_square: raise NonSquareMatrixError("Matrix must be square.") if not self.is_symmetric(): raise ValueError("Matrix must be symmetric.") return self._cholesky()
[docs] def condition_number(self): """Returns the condition number of a matrix. This is the maximum singular value divided by the minimum singular value Examples ======== >>> from sympy import Matrix, S >>> A = Matrix([[1, 0, 0], [0, 10, 0], [0, 0, S.One/10]]) >>> A.condition_number() 100 See Also ======== singular_values """ if not self: return S.Zero singularvalues = self.singular_values() return Max(*singularvalues) / Min(*singularvalues)
[docs] def copy(self): """ Returns the copy of a matrix. Examples ======== >>> from sympy import Matrix >>> A = Matrix(2, 2, [1, 2, 3, 4]) >>> A.copy() Matrix([ [1, 2], [3, 4]]) """ return self._new(self.rows, self.cols, self._mat)
[docs] def cross(self, b): """Return the cross product of `self` and `b` relaxing the condition of compatible dimensions: if each has 3 elements, a matrix of the same type and shape as `self` will be returned. If `b` has the same shape as `self` then common identities for the cross product (like `a x b = - b x a`) will hold. See Also ======== dot multiply multiply_elementwise """ if not is_sequence(b): raise TypeError( "`b` must be an ordered iterable or Matrix, not %s." % type(b)) if not (self.rows * self.cols == b.rows * b.cols == 3): raise ShapeError("Dimensions incorrect for cross product: %s x %s" % ((self.rows, self.cols), (b.rows, b.cols))) else: return self._new(self.rows, self.cols, ( (self[1] * b[2] - self[2] * b[1]), (self[2] * b[0] - self[0] * b[2]), (self[0] * b[1] - self[1] * b[0])))
@property def D(self): """Return Dirac conjugate (if self.rows == 4). Examples ======== >>> from sympy import Matrix, I, eye >>> m = Matrix((0, 1 + I, 2, 3)) >>> m.D Matrix([[0, 1 - I, -2, -3]]) >>> m = (eye(4) + I*eye(4)) >>> m[0, 3] = 2 >>> m.D Matrix([ [1 - I, 0, 0, 0], [ 0, 1 - I, 0, 0], [ 0, 0, -1 + I, 0], [ 2, 0, 0, -1 + I]]) If the matrix does not have 4 rows an AttributeError will be raised because this property is only defined for matrices with 4 rows. >>> Matrix(eye(2)).D Traceback (most recent call last): ... AttributeError: Matrix has no attribute D. See Also ======== conjugate: By-element conjugation H: Hermite conjugation """ from sympy.physics.matrices import mgamma if self.rows != 4: # In Python 3.2, properties can only return an AttributeError # so we can't raise a ShapeError -- see commit which added the # first line of this inline comment. Also, there is no need # for a message since MatrixBase will raise the AttributeError raise AttributeError return self.H * mgamma(0)
[docs] def diagonal_solve(self, rhs): """Solves Ax = B efficiently, where A is a diagonal Matrix, with non-zero diagonal entries. Examples ======== >>> from sympy.matrices import Matrix, eye >>> A = eye(2)*2 >>> B = Matrix([[1, 2], [3, 4]]) >>> A.diagonal_solve(B) == B/2 True See Also ======== lower_triangular_solve upper_triangular_solve gauss_jordan_solve cholesky_solve LDLsolve LUsolve QRsolve pinv_solve """ if not self.is_diagonal: raise TypeError("Matrix should be diagonal") if rhs.rows != self.rows: raise TypeError("Size mis-match") return self._diagonal_solve(rhs)
[docs] def dot(self, b): """Return the dot product of Matrix self and b relaxing the condition of compatible dimensions: if either the number of rows or columns are the same as the length of b then the dot product is returned. If self is a row or column vector, a scalar is returned. Otherwise, a list of results is returned (and in that case the number of columns in self must match the length of b). Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> v = [1, 1, 1] >>> M.row(0).dot(v) 6 >>> M.col(0).dot(v) 12 >>> M.dot(v) [6, 15, 24] See Also ======== cross multiply multiply_elementwise """ from .dense import Matrix if not isinstance(b, MatrixBase): if is_sequence(b): if len(b) != self.cols and len(b) != self.rows: raise ShapeError( "Dimensions incorrect for dot product: %s, %s" % ( self.shape, len(b))) return self.dot(Matrix(b)) else: raise TypeError( "`b` must be an ordered iterable or Matrix, not %s." % type(b)) mat = self if mat.cols == b.rows: if b.cols != 1: mat = mat.T b = b.T prod = flatten((mat * b).tolist()) if len(prod) == 1: return prod[0] return prod if mat.cols == b.cols: return mat.dot(b.T) elif mat.rows == b.rows: return mat.T.dot(b) else: raise ShapeError("Dimensions incorrect for dot product: %s, %s" % ( self.shape, b.shape))
[docs] def dual(self): """Returns the dual of a matrix, which is: `(1/2)*levicivita(i, j, k, l)*M(k, l)` summed over indices `k` and `l` Since the levicivita method is anti_symmetric for any pairwise exchange of indices, the dual of a symmetric matrix is the zero matrix. Strictly speaking the dual defined here assumes that the 'matrix' `M` is a contravariant anti_symmetric second rank tensor, so that the dual is a covariant second rank tensor. """ from sympy import LeviCivita from sympy.matrices import zeros M, n = self[:, :], self.rows work = zeros(n) if self.is_symmetric(): return work for i in range(1, n): for j in range(1, n): acum = 0 for k in range(1, n): acum += LeviCivita(i, j, 0, k) * M[0, k] work[i, j] = acum work[j, i] = -acum for l in range(1, n): acum = 0 for a in range(1, n): for b in range(1, n): acum += LeviCivita(0, l, a, b) * M[a, b] acum /= 2 work[0, l] = -acum work[l, 0] = acum return work
[docs] def exp(self): """Return the exponentiation of a square matrix.""" if not self.is_square: raise NonSquareMatrixError( "Exponentiation is valid only for square matrices") try: P, J = self.jordan_form() cells = J.get_diag_blocks() except MatrixError: raise NotImplementedError( "Exponentiation is implemented only for matrices for which the Jordan normal form can be computed") def _jblock_exponential(b): # This function computes the matrix exponential for one single Jordan block nr = b.rows l = b[0, 0] if nr == 1: res = exp(l) else: from sympy import eye # extract the diagonal part d = b[0, 0] * eye(nr) # and the nilpotent part n = b - d # compute its exponential nex = eye(nr) for i in range(1, nr): nex = nex + n ** i / factorial(i) # combine the two parts res = exp(b[0, 0]) * nex return (res) blocks = list(map(_jblock_exponential, cells)) from sympy.matrices import diag eJ = diag(*blocks) # n = self.rows ret = P * eJ * P.inv() return type(self)(ret)
[docs] def gauss_jordan_solve(self, b, freevar=False): """ Solves Ax = b using Gauss Jordan elimination. There may be zero, one, or infinite solutions. If one solution exists, it will be returned. If infinite solutions exist, it will be returned parametrically. If no solutions exist, It will throw ValueError. Parameters ========== b : Matrix The right hand side of the equation to be solved for. Must have the same number of rows as matrix A. freevar : List If the system is underdetermined (e.g. A has more columns than rows), infinite solutions are possible, in terms of an arbitrary values of free variables. Then the index of the free variables in the solutions (column Matrix) will be returned by freevar, if the flag `freevar` is set to `True`. Returns ======= x : Matrix The matrix that will satisfy Ax = B. Will have as many rows as matrix A has columns, and as many columns as matrix B. params : Matrix If the system is underdetermined (e.g. A has more columns than rows), infinite solutions are possible, in terms of an arbitrary parameters. These arbitrary parameters are returned as params Matrix. Examples ======== >>> from sympy import Matrix >>> A = Matrix([[1, 2, 1, 1], [1, 2, 2, -1], [2, 4, 0, 6]]) >>> b = Matrix([7, 12, 4]) >>> sol, params = A.gauss_jordan_solve(b) >>> sol Matrix([ [-2*_tau0 - 3*_tau1 + 2], [ _tau0], [ 2*_tau1 + 5], [ _tau1]]) >>> params Matrix([ [_tau0], [_tau1]]) >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]]) >>> b = Matrix([3, 6, 9]) >>> sol, params = A.gauss_jordan_solve(b) >>> sol Matrix([ [-1], [ 2], [ 0]]) >>> params Matrix(0, 1, []) See Also ======== lower_triangular_solve upper_triangular_solve cholesky_solve diagonal_solve LDLsolve LUsolve QRsolve pinv References ========== .. [1] http://en.wikipedia.org/wiki/Gaussian_elimination """ from sympy.matrices import Matrix, zeros aug = self.hstack(self.copy(), b.copy()) row, col = aug[:, :-1].shape # solve by reduced row echelon form A, pivots = aug.rref(simplify=True) A, v = A[:, :-1], A[:, -1] pivots = list(filter(lambda p: p < col, pivots)) rank = len(pivots) # Bring to block form permutation = Matrix(range(col)).T A = A.vstack(A, permutation) for i, c in enumerate(pivots): A.col_swap(i, c) A, permutation = A[:-1, :], A[-1, :] # check for existence of solutions # rank of aug Matrix should be equal to rank of coefficient matrix if not v[rank:, 0].is_zero: raise ValueError("Linear system has no solution") # Get index of free symbols (free parameters) free_var_index = permutation[ len(pivots):] # non-pivots columns are free variables # Free parameters dummygen = numbered_symbols("tau", Dummy) tau = Matrix([next(dummygen) for k in range(col - rank)]).reshape( col - rank, 1) # Full parametric solution V = A[:rank, rank:] vt = v[:rank, 0] free_sol = tau.vstack(vt - V * tau, tau) # Undo permutation sol = zeros(col, 1) for k, v in enumerate(free_sol): sol[permutation[k], 0] = v if freevar: return sol, tau, free_var_index else: return sol, tau
[docs] def inv_mod(self, m): r""" Returns the inverse of the matrix `K` (mod `m`), if it exists. Method to find the matrix inverse of `K` (mod `m`) implemented in this function: * Compute `\mathrm{adj}(K) = \mathrm{cof}(K)^t`, the adjoint matrix of `K`. * Compute `r = 1/\mathrm{det}(K) \pmod m`. * `K^{-1} = r\cdot \mathrm{adj}(K) \pmod m`. Examples ======== >>> from sympy import Matrix >>> A = Matrix(2, 2, [1, 2, 3, 4]) >>> A.inv_mod(5) Matrix([ [3, 1], [4, 2]]) >>> A.inv_mod(3) Matrix([ [1, 1], [0, 1]]) """ from sympy.ntheory import totient if not self.is_square: raise NonSquareMatrixError() N = self.cols phi = totient(m) det_K = self.det() if gcd(det_K, m) != 1: raise ValueError('Matrix is not invertible (mod %d)' % m) det_inv = pow(int(det_K), int(phi - 1), int(m)) K_adj = self.adjugate() K_inv = self.__class__(N, N, [det_inv * K_adj[i, j] % m for i in range(N) for j in range(N)]) return K_inv
[docs] def inverse_ADJ(self, iszerofunc=_iszero): """Calculates the inverse using the adjugate matrix and a determinant. See Also ======== inv inverse_LU inverse_GE """ if not self.is_square: raise NonSquareMatrixError("A Matrix must be square to invert.") d = self.det(method='berkowitz') zero = d.equals(0) if zero is None: # if equals() can't decide, will rref be able to? ok = self.rref(simplify=True)[0] zero = any(iszerofunc(ok[j, j]) for j in range(ok.rows)) if zero: raise ValueError("Matrix det == 0; not invertible.") return self.adjugate() / d
[docs] def inverse_GE(self, iszerofunc=_iszero): """Calculates the inverse using Gaussian elimination. See Also ======== inv inverse_LU inverse_ADJ """ from .dense import Matrix if not self.is_square: raise NonSquareMatrixError("A Matrix must be square to invert.") big = Matrix.hstack(self.as_mutable(), Matrix.eye(self.rows)) red = big.rref(iszerofunc=iszerofunc, simplify=True)[0] if any(iszerofunc(red[j, j]) for j in range(red.rows)): raise ValueError("Matrix det == 0; not invertible.") return self._new(red[:, big.rows:])
[docs] def inverse_LU(self, iszerofunc=_iszero): """Calculates the inverse using LU decomposition. See Also ======== inv inverse_GE inverse_ADJ """ if not self.is_square: raise NonSquareMatrixError() ok = self.rref(simplify=True)[0] if any(iszerofunc(ok[j, j]) for j in range(ok.rows)): raise ValueError("Matrix det == 0; not invertible.") return self.LUsolve(self.eye(self.rows), iszerofunc=_iszero)
[docs] def inv(self, method=None, **kwargs): """ Return the inverse of a matrix. CASE 1: If the matrix is a dense matrix. Return the matrix inverse using the method indicated (default is Gauss elimination). Parameters ========== method : ('GE', 'LU', or 'ADJ') Notes ===== According to the ``method`` keyword, it calls the appropriate method: GE .... inverse_GE(); default LU .... inverse_LU() ADJ ... inverse_ADJ() See Also ======== inverse_LU inverse_GE inverse_ADJ Raises ------ ValueError If the determinant of the matrix is zero. CASE 2: If the matrix is a sparse matrix. Return the matrix inverse using Cholesky or LDL (default). kwargs ====== method : ('CH', 'LDL') Notes ===== According to the ``method`` keyword, it calls the appropriate method: LDL ... inverse_LDL(); default CH .... inverse_CH() Raises ------ ValueError If the determinant of the matrix is zero. """ if not self.is_square: raise NonSquareMatrixError() if method is not None: kwargs['method'] = method return self._eval_inverse(**kwargs)
[docs] def is_nilpotent(self): """Checks if a matrix is nilpotent. A matrix B is nilpotent if for some integer k, B**k is a zero matrix. Examples ======== >>> from sympy import Matrix >>> a = Matrix([[0, 0, 0], [1, 0, 0], [1, 1, 0]]) >>> a.is_nilpotent() True >>> a = Matrix([[1, 0, 1], [1, 0, 0], [1, 1, 0]]) >>> a.is_nilpotent() False """ if not self: return True if not self.is_square: raise NonSquareMatrixError( "Nilpotency is valid only for square matrices") x = Dummy('x') if self.charpoly(x).args[0] == x ** self.rows: return True return False
[docs] def key2bounds(self, keys): """Converts a key with potentially mixed types of keys (integer and slice) into a tuple of ranges and raises an error if any index is out of self's range. See Also ======== key2ij """ islice, jslice = [isinstance(k, slice) for k in keys] if islice: if not self.rows: rlo = rhi = 0 else: rlo, rhi = keys[0].indices(self.rows)[:2] else: rlo = a2idx(keys[0], self.rows) rhi = rlo + 1 if jslice: if not self.cols: clo = chi = 0 else: clo, chi = keys[1].indices(self.cols)[:2] else: clo = a2idx(keys[1], self.cols) chi = clo + 1 return rlo, rhi, clo, chi
[docs] def key2ij(self, key): """Converts key into canonical form, converting integers or indexable items into valid integers for self's range or returning slices unchanged. See Also ======== key2bounds """ if is_sequence(key): if not len(key) == 2: raise TypeError('key must be a sequence of length 2') return [a2idx(i, n) if not isinstance(i, slice) else i for i, n in zip(key, self.shape)] elif isinstance(key, slice): return key.indices(len(self))[:2] else: return divmod(a2idx(key, len(self)), self.cols)
[docs] def LDLdecomposition(self): """Returns the LDL Decomposition (L, D) of matrix A, such that L * D * L.T == A This method eliminates the use of square root. Further this ensures that all the diagonal entries of L are 1. A must be a square, symmetric, positive-definite and non-singular matrix. Examples ======== >>> from sympy.matrices import Matrix, eye >>> A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11))) >>> L, D = A.LDLdecomposition() >>> L Matrix([ [ 1, 0, 0], [ 3/5, 1, 0], [-1/5, 1/3, 1]]) >>> D Matrix([ [25, 0, 0], [ 0, 9, 0], [ 0, 0, 9]]) >>> L * D * L.T * A.inv() == eye(A.rows) True See Also ======== cholesky LUdecomposition QRdecomposition """ if not self.is_square: raise NonSquareMatrixError("Matrix must be square.") if not self.is_symmetric(): raise ValueError("Matrix must be symmetric.") return self._LDLdecomposition()
[docs] def LDLsolve(self, rhs): """Solves Ax = B using LDL decomposition, for a general square and non-singular matrix. For a non-square matrix with rows > cols, the least squares solution is returned. Examples ======== >>> from sympy.matrices import Matrix, eye >>> A = eye(2)*2 >>> B = Matrix([[1, 2], [3, 4]]) >>> A.LDLsolve(B) == B/2 True See Also ======== LDLdecomposition lower_triangular_solve upper_triangular_solve gauss_jordan_solve cholesky_solve diagonal_solve LUsolve QRsolve pinv_solve """ if self.is_symmetric(): L, D = self.LDLdecomposition() elif self.rows >= self.cols: L, D = (self.T * self).LDLdecomposition() rhs = self.T * rhs else: raise NotImplementedError('Under-determined System. ' 'Try M.gauss_jordan_solve(rhs)') Y = L._lower_triangular_solve(rhs) Z = D._diagonal_solve(Y) return (L.T)._upper_triangular_solve(Z)
[docs] def lower_triangular_solve(self, rhs): """Solves Ax = B, where A is a lower triangular matrix. See Also ======== upper_triangular_solve gauss_jordan_solve cholesky_solve diagonal_solve LDLsolve LUsolve QRsolve pinv_solve """ if not self.is_square: raise NonSquareMatrixError("Matrix must be square.") if rhs.rows != self.rows: raise ShapeError("Matrices size mismatch.") if not self.is_lower: raise ValueError("Matrix must be lower triangular.") return self._lower_triangular_solve(rhs)
[docs] def LUdecomposition(self, iszerofunc=_iszero, simpfunc=None, rankcheck=False): """Returns (L, U, perm) where L is a lower triangular matrix with unit diagonal, U is an upper triangular matrix, and perm is a list of row swap index pairs. If A is the original matrix, then A = (L*U).permuteBkwd(perm), and the row permutation matrix P such that P*A = L*U can be computed by P=eye(A.row).permuteFwd(perm). See documentation for LUCombined for details about the keyword argument rankcheck, iszerofunc, and simpfunc. Examples ======== >>> from sympy import Matrix >>> a = Matrix([[4, 3], [6, 3]]) >>> L, U, _ = a.LUdecomposition() >>> L Matrix([ [ 1, 0], [3/2, 1]]) >>> U Matrix([ [4, 3], [0, -3/2]]) See Also ======== cholesky LDLdecomposition QRdecomposition LUdecomposition_Simple LUdecompositionFF LUsolve """ combined, p = self.LUdecomposition_Simple(iszerofunc=iszerofunc, simpfunc=simpfunc, rankcheck=rankcheck) # L is lower triangular self.rows x self.rows # U is upper triangular self.rows x self.cols # L has unit diagonal. For each column in combined, the subcolumn # below the diagonal of combined is shared by L. # If L has more columns than combined, then the remaining subcolumns # below the diagonal of L are zero. # The upper triangular portion of L and combined are equal. def entry_L(i, j): if i < j: # Super diagonal entry return S.Zero elif i == j: return S.One elif j < combined.cols: return combined[i, j] # Subdiagonal entry of L with no corresponding # entry in combined return S.Zero def entry_U(i, j): return S.Zero if i > j else combined[i, j] L = self._new(combined.rows, combined.rows, entry_L) U = self._new(combined.rows, combined.cols, entry_U) return L, U, p
[docs] def LUdecomposition_Simple(self, iszerofunc=_iszero, simpfunc=None, rankcheck=False): """Compute an lu decomposition of m x n matrix A, where P*A = L*U * L is m x m lower triangular with unit diagonal * U is m x n upper triangular * P is an m x m permutation matrix Returns an m x n matrix lu, and an m element list perm where each element of perm is a pair of row exchange indices. The factors L and U are stored in lu as follows: The subdiagonal elements of L are stored in the subdiagonal elements of lu, that is lu[i, j] = L[i, j] whenever i > j. The elements on the diagonal of L are all 1, and are not explicitly stored. U is stored in the upper triangular portion of lu, that is lu[i ,j] = U[i, j] whenever i <= j. The output matrix can be visualized as: Matrix([ [u, u, u, u], [l, u, u, u], [l, l, u, u], [l, l, l, u]]) where l represents a subdiagonal entry of the L factor, and u represents an entry from the upper triangular entry of the U factor. perm is a list row swap index pairs such that if A is the original matrix, then A = (L*U).permuteBkwd(perm), and the row permutation matrix P such that P*A = L*U can be computed by soP=eye(A.row).permuteFwd(perm). The keyword argument rankcheck determines if this function raises a ValueError when passed a matrix whose rank is strictly less than min(num rows, num cols). The default behavior is to decompose a rank deficient matrix. Pass rankcheck=True to raise a ValueError instead. (This mimics the previous behavior of this function). The keyword arguments iszerofunc and simpfunc are used by the pivot search algorithm. iszerofunc is a callable that returns a boolean indicating if its input is zero, or None if it cannot make the determination. simpfunc is a callable that simplifies its input. The default is simpfunc=None, which indicate that the pivot search algorithm should not attempt to simplify any candidate pivots. If simpfunc fails to simplify its input, then it must return its input instead of a copy. When a matrix contains symbolic entries, the pivot search algorithm differs from the case where every entry can be categorized as zero or nonzero. The algorithm searches column by column through the submatrix whose top left entry coincides with the pivot position. If it exists, the pivot is the first entry in the current search column that iszerofunc guarantees is nonzero. If no such candidate exists, then each candidate pivot is simplified if simpfunc is not None. The search is repeated, with the difference that a candidate may be the pivot if `iszerofunc()` cannot guarantee that it is nonzero. In the second search the pivot is the first candidate that iszerofunc can guarantee is nonzero. If no such candidate exists, then the pivot is the first candidate for which iszerofunc returns None. If no such candidate exists, then the search is repeated in the next column to the right. The pivot search algorithm differs from the one in `rref()`, which relies on `_find_reasonable_pivot()`. Future versions of `LUdecomposition_simple()` may use `_find_reasonable_pivot()`. See Also ======== LUdecomposition LUdecompositionFF LUsolve """ if rankcheck: # https://github.com/sympy/sympy/issues/9796 pass if self.rows == 0 or self.cols == 0: # Define LU decomposition of a matrix with no entries as a matrix # of the same dimensions with all zero entries. return self.zeros(self.rows, self.cols), [] lu = self.as_mutable() row_swaps = [] pivot_col = 0 for pivot_row in range(0, lu.rows - 1): # Search for pivot. Prefer entry that iszeropivot determines # is nonzero, over entry that iszeropivot cannot guarantee # is zero. # XXX `_find_reasonable_pivot` uses slow zero testing. Blocked by bug #10279 # Future versions of LUdecomposition_simple can pass iszerofunc and simpfunc # to _find_reasonable_pivot(). # In pass 3 of _find_reasonable_pivot(), the predicate in `if x.equals(S.Zero):` # calls sympy.simplify(), and not the simplification function passed in via # the keyword argument simpfunc. iszeropivot = True while pivot_col != self.cols and iszeropivot: sub_col = (lu[r, pivot_col] for r in range(pivot_row, self.rows)) pivot_row_offset, pivot_value, is_assumed_non_zero, ind_simplified_pairs =\ _find_reasonable_pivot_naive(sub_col, iszerofunc, simpfunc) iszeropivot = pivot_value is None if iszeropivot: # All candidate pivots in this column are zero. # Proceed to next column. pivot_col += 1 if rankcheck and pivot_col != pivot_row: # All entries including and below the pivot position are # zero, which indicates that the rank of the matrix is # strictly less than min(num rows, num cols) # Mimic behavior of previous implementation, by throwing a # ValueError. raise ValueError("Rank of matrix is strictly less than" " number of rows or columns." " Pass keyword argument" " rankcheck=False to compute" " the LU decomposition of this matrix.") candidate_pivot_row = None if pivot_row_offset is None else pivot_row + pivot_row_offset if candidate_pivot_row is None and iszeropivot: # If candidate_pivot_row is None and iszeropivot is True # after pivot search has completed, then the submatrix # below and to the right of (pivot_row, pivot_col) is # all zeros, indicating that Gaussian elimination is # complete. return lu, row_swaps # Update entries simplified during pivot search. for offset, val in ind_simplified_pairs: lu[pivot_row + offset, pivot_col] = val if pivot_row != candidate_pivot_row: # Row swap book keeping: # Record which rows were swapped. # Update stored portion of L factor by multiplying L on the # left and right with the current permutation. # Swap rows of U. row_swaps.append([pivot_row, candidate_pivot_row]) # Update L. lu[pivot_row, 0:pivot_row], lu[candidate_pivot_row, 0:pivot_row] = \ lu[candidate_pivot_row, 0:pivot_row], lu[pivot_row, 0:pivot_row] # Swap pivot row of U with candidate pivot row. lu[pivot_row, pivot_col:lu.cols], lu[candidate_pivot_row, pivot_col:lu.cols] = \ lu[candidate_pivot_row, pivot_col:lu.cols], lu[pivot_row, pivot_col:lu.cols] # Introduce zeros below the pivot by adding a multiple of the # pivot row to a row under it, and store the result in the # row under it. # Only entries in the target row whose index is greater than # start_col may be nonzero. start_col = pivot_col + 1 for row in range(pivot_row + 1, lu.rows): # Store factors of L in the subcolumn below # (pivot_row, pivot_row). lu[row, pivot_row] =\ lu[row, pivot_col]/lu[pivot_row, pivot_col] # Form the linear combination of the pivot row and the current # row below the pivot row that zeros the entries below the pivot. # Employing slicing instead of a loop here raises # NotImplementedError: Cannot add Zero to MutableSparseMatrix # in sympy/matrices/tests/test_sparse.py. # c = pivot_row + 1 if pivot_row == pivot_col else pivot_col for c in range(start_col, lu.cols): lu[row, c] = lu[row, c] - lu[row, pivot_row]*lu[pivot_row, c] if pivot_row != pivot_col: # matrix rank < min(num rows, num cols), # so factors of L are not stored directly below the pivot. # These entries are zero by construction, so don't bother # computing them. for row in range(pivot_row + 1, lu.rows): lu[row, pivot_col] = S.Zero pivot_col += 1 if pivot_col == lu.cols: # All candidate pivots are zero implies that Gaussian # elimination is complete. return lu, row_swaps return lu, row_swaps
[docs] def LUdecompositionFF(self): """Compute a fraction-free LU decomposition. Returns 4 matrices P, L, D, U such that PA = L D**-1 U. If the elements of the matrix belong to some integral domain I, then all elements of L, D and U are guaranteed to belong to I. **Reference** - W. Zhou & D.J. Jeffrey, "Fraction-free matrix factors: new forms for LU and QR factors". Frontiers in Computer Science in China, Vol 2, no. 1, pp. 67-80, 2008. See Also ======== LUdecomposition LUdecomposition_Simple LUsolve """ from sympy.matrices import SparseMatrix zeros = SparseMatrix.zeros eye = SparseMatrix.eye n, m = self.rows, self.cols U, L, P = self.as_mutable(), eye(n), eye(n) DD = zeros(n, n) oldpivot = 1 for k in range(n - 1): if U[k, k] == 0: for kpivot in range(k + 1, n): if U[kpivot, k]: break else: raise ValueError("Matrix is not full rank") U[k, k:], U[kpivot, k:] = U[kpivot, k:], U[k, k:] L[k, :k], L[kpivot, :k] = L[kpivot, :k], L[k, :k] P[k, :], P[kpivot, :] = P[kpivot, :], P[k, :] L[k, k] = Ukk = U[k, k] DD[k, k] = oldpivot * Ukk for i in range(k + 1, n): L[i, k] = Uik = U[i, k] for j in range(k + 1, m): U[i, j] = (Ukk * U[i, j] - U[k, j] * Uik) / oldpivot U[i, k] = 0 oldpivot = Ukk DD[n - 1, n - 1] = oldpivot return P, L, DD, U
[docs] def LUsolve(self, rhs, iszerofunc=_iszero): """Solve the linear system Ax = rhs for x where A = self. This is for symbolic matrices, for real or complex ones use mpmath.lu_solve or mpmath.qr_solve. See Also ======== lower_triangular_solve upper_triangular_solve gauss_jordan_solve cholesky_solve diagonal_solve LDLsolve QRsolve pinv_solve LUdecomposition """ if rhs.rows != self.rows: raise ShapeError( "`self` and `rhs` must have the same number of rows.") A, perm = self.LUdecomposition_Simple(iszerofunc=_iszero) n = self.rows b = rhs.permute_rows(perm).as_mutable() # forward substitution, all diag entries are scaled to 1 for i in range(n): for j in range(i): scale = A[i, j] b.zip_row_op(i, j, lambda x, y: x - y * scale) # backward substitution for i in range(n - 1, -1, -1): for j in range(i + 1, n): scale = A[i, j] b.zip_row_op(i, j, lambda x, y: x - y * scale) scale = A[i, i] b.row_op(i, lambda x, _: x / scale) return rhs.__class__(b)
[docs] def multiply(self, b): """Returns self*b See Also ======== dot cross multiply_elementwise """ return self * b
[docs] def normalized(self): """Return the normalized version of ``self``. See Also ======== norm """ if self.rows != 1 and self.cols != 1: raise ShapeError("A Matrix must be a vector to normalize.") norm = self.norm() out = self.applyfunc(lambda i: i / norm) return out
[docs] def norm(self, ord=None): """Return the Norm of a Matrix or Vector. In the simplest case this is the geometric size of the vector Other norms can be specified by the ord parameter ===== ============================ ========================== ord norm for matrices norm for vectors ===== ============================ ========================== None Frobenius norm 2-norm 'fro' Frobenius norm - does not exist inf -- max(abs(x)) -inf -- min(abs(x)) 1 -- as below -1 -- as below 2 2-norm (largest sing. value) as below -2 smallest singular value as below other - does not exist sum(abs(x)**ord)**(1./ord) ===== ============================ ========================== Examples ======== >>> from sympy import Matrix, Symbol, trigsimp, cos, sin, oo >>> x = Symbol('x', real=True) >>> v = Matrix([cos(x), sin(x)]) >>> trigsimp( v.norm() ) 1 >>> v.norm(10) (sin(x)**10 + cos(x)**10)**(1/10) >>> A = Matrix([[1, 1], [1, 1]]) >>> A.norm(2)# Spectral norm (max of |Ax|/|x| under 2-vector-norm) 2 >>> A.norm(-2) # Inverse spectral norm (smallest singular value) 0 >>> A.norm() # Frobenius Norm 2 >>> Matrix([1, -2]).norm(oo) 2 >>> Matrix([-1, 2]).norm(-oo) 1 See Also ======== normalized """ # Row or Column Vector Norms vals = list(self.values()) or [0] if self.rows == 1 or self.cols == 1: if ord == 2 or ord is None: # Common case sqrt(<x, x>) return sqrt(Add(*(abs(i) ** 2 for i in vals))) elif ord == 1: # sum(abs(x)) return Add(*(abs(i) for i in vals)) elif ord == S.Infinity: # max(abs(x)) return Max(*[abs(i) for i in vals]) elif ord == S.NegativeInfinity: # min(abs(x)) return Min(*[abs(i) for i in vals]) # Otherwise generalize the 2-norm, Sum(x_i**ord)**(1/ord) # Note that while useful this is not mathematically a norm try: return Pow(Add(*(abs(i) ** ord for i in vals)), S(1) / ord) except (NotImplementedError, TypeError): raise ValueError("Expected order to be Number, Symbol, oo") # Matrix Norms else: if ord == 2: # Spectral Norm # Maximum singular value return Max(*self.singular_values()) elif ord == -2: # Minimum singular value return Min(*self.singular_values()) elif (ord is None or isinstance(ord, string_types) and ord.lower() in ['f', 'fro', 'frobenius', 'vector']): # Reshape as vector and send back to norm function return self.vec().norm(ord=2) else: raise NotImplementedError("Matrix Norms under development")
[docs] def pinv_solve(self, B, arbitrary_matrix=None): """Solve Ax = B using the Moore-Penrose pseudoinverse. There may be zero, one, or infinite solutions. If one solution exists, it will be returned. If infinite solutions exist, one will be returned based on the value of arbitrary_matrix. If no solutions exist, the least-squares solution is returned. Parameters ========== B : Matrix The right hand side of the equation to be solved for. Must have the same number of rows as matrix A. arbitrary_matrix : Matrix If the system is underdetermined (e.g. A has more columns than rows), infinite solutions are possible, in terms of an arbitrary matrix. This parameter may be set to a specific matrix to use for that purpose; if so, it must be the same shape as x, with as many rows as matrix A has columns, and as many columns as matrix B. If left as None, an appropriate matrix containing dummy symbols in the form of ``wn_m`` will be used, with n and m being row and column position of each symbol. Returns ======= x : Matrix The matrix that will satisfy Ax = B. Will have as many rows as matrix A has columns, and as many columns as matrix B. Examples ======== >>> from sympy import Matrix >>> A = Matrix([[1, 2, 3], [4, 5, 6]]) >>> B = Matrix([7, 8]) >>> A.pinv_solve(B) Matrix([ [ _w0_0/6 - _w1_0/3 + _w2_0/6 - 55/18], [-_w0_0/3 + 2*_w1_0/3 - _w2_0/3 + 1/9], [ _w0_0/6 - _w1_0/3 + _w2_0/6 + 59/18]]) >>> A.pinv_solve(B, arbitrary_matrix=Matrix([0, 0, 0])) Matrix([ [-55/18], [ 1/9], [ 59/18]]) See Also ======== lower_triangular_solve upper_triangular_solve gauss_jordan_solve cholesky_solve diagonal_solve LDLsolve LUsolve QRsolve pinv Notes ===== This may return either exact solutions or least squares solutions. To determine which, check ``A * A.pinv() * B == B``. It will be True if exact solutions exist, and False if only a least-squares solution exists. Be aware that the left hand side of that equation may need to be simplified to correctly compare to the right hand side. References ========== .. [1] https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#Obtaining_all_solutions_of_a_linear_system """ from sympy.matrices import eye A = self A_pinv = self.pinv() if arbitrary_matrix is None: rows, cols = A.cols, B.cols w = symbols('w:{0}_:{1}'.format(rows, cols), cls=Dummy) arbitrary_matrix = self.__class__(cols, rows, w).T return A_pinv * B + (eye(A.cols) - A_pinv * A) * arbitrary_matrix
[docs] def pinv(self): """Calculate the Moore-Penrose pseudoinverse of the matrix. The Moore-Penrose pseudoinverse exists and is unique for any matrix. If the matrix is invertible, the pseudoinverse is the same as the inverse. Examples ======== >>> from sympy import Matrix >>> Matrix([[1, 2, 3], [4, 5, 6]]).pinv() Matrix([ [-17/18, 4/9], [ -1/9, 1/9], [ 13/18, -2/9]]) See Also ======== inv pinv_solve References ========== .. [1] https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse """ A = self AH = self.H # Trivial case: pseudoinverse of all-zero matrix is its transpose. if A.is_zero: return AH try: if self.rows >= self.cols: return (AH * A).inv() * AH else: return AH * (A * AH).inv() except ValueError: # Matrix is not full rank, so A*AH cannot be inverted. raise NotImplementedError('Rank-deficient matrices are not yet ' 'supported.')
[docs] def print_nonzero(self, symb="X"): """Shows location of non-zero entries for fast shape lookup. Examples ======== >>> from sympy.matrices import Matrix, eye >>> m = Matrix(2, 3, lambda i, j: i*3+j) >>> m Matrix([ [0, 1, 2], [3, 4, 5]]) >>> m.print_nonzero() [ XX] [XXX] >>> m = eye(4) >>> m.print_nonzero("x") [x ] [ x ] [ x ] [ x] """ s = [] for i in range(self.rows): line = [] for j in range(self.cols): if self[i, j] == 0: line.append(" ") else: line.append(str(symb)) s.append("[%s]" % ''.join(line)) print('\n'.join(s))
[docs] def project(self, v): """Return the projection of ``self`` onto the line containing ``v``. Examples ======== >>> from sympy import Matrix, S, sqrt >>> V = Matrix([sqrt(3)/2, S.Half]) >>> x = Matrix([[1, 0]]) >>> V.project(x) Matrix([[sqrt(3)/2, 0]]) >>> V.project(-x) Matrix([[sqrt(3)/2, 0]]) """ return v * (self.dot(v) / v.dot(v))
[docs] def QRdecomposition(self): """Return Q, R where A = Q*R, Q is orthogonal and R is upper triangular. Examples ======== This is the example from wikipedia: >>> from sympy import Matrix >>> A = Matrix([[12, -51, 4], [6, 167, -68], [-4, 24, -41]]) >>> Q, R = A.QRdecomposition() >>> Q Matrix([ [ 6/7, -69/175, -58/175], [ 3/7, 158/175, 6/175], [-2/7, 6/35, -33/35]]) >>> R Matrix([ [14, 21, -14], [ 0, 175, -70], [ 0, 0, 35]]) >>> A == Q*R True QR factorization of an identity matrix: >>> A = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> Q, R = A.QRdecomposition() >>> Q Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> R Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) See Also ======== cholesky LDLdecomposition LUdecomposition QRsolve """ cls = self.__class__ mat = self.as_mutable() if not mat.rows >= mat.cols: raise MatrixError( "The number of rows must be greater than columns") n = mat.rows m = mat.cols rank = n row_reduced = mat.rref()[0] for i in range(row_reduced.rows): if row_reduced.row(i).norm() == 0: rank -= 1 if not rank == mat.cols: raise MatrixError("The rank of the matrix must match the columns") Q, R = mat.zeros(n, m), mat.zeros(m) for j in range(m): # for each column vector tmp = mat[:, j] # take original v for i in range(j): # subtract the project of mat on new vector tmp -= Q[:, i] * mat[:, j].dot(Q[:, i]) tmp.expand() # normalize it R[j, j] = tmp.norm() Q[:, j] = tmp / R[j, j] if Q[:, j].norm() != 1: raise NotImplementedError( "Could not normalize the vector %d." % j) for i in range(j): R[i, j] = Q[:, i].dot(mat[:, j]) return cls(Q), cls(R)
[docs] def QRsolve(self, b): """Solve the linear system 'Ax = b'. 'self' is the matrix 'A', the method argument is the vector 'b'. The method returns the solution vector 'x'. If 'b' is a matrix, the system is solved for each column of 'b' and the return value is a matrix of the same shape as 'b'. This method is slower (approximately by a factor of 2) but more stable for floating-point arithmetic than the LUsolve method. However, LUsolve usually uses an exact arithmetic, so you don't need to use QRsolve. This is mainly for educational purposes and symbolic matrices, for real (or complex) matrices use mpmath.qr_solve. See Also ======== lower_triangular_solve upper_triangular_solve gauss_jordan_solve cholesky_solve diagonal_solve LDLsolve LUsolve pinv_solve QRdecomposition """ Q, R = self.as_mutable().QRdecomposition() y = Q.T * b # back substitution to solve R*x = y: # We build up the result "backwards" in the vector 'x' and reverse it # only in the end. x = [] n = R.rows for j in range(n - 1, -1, -1): tmp = y[j, :] for k in range(j + 1, n): tmp -= R[j, k] * x[n - 1 - k] x.append(tmp / R[j, j]) return self._new([row._mat for row in reversed(x)])
[docs] def solve_least_squares(self, rhs, method='CH'): """Return the least-square fit to the data. By default the cholesky_solve routine is used (method='CH'); other methods of matrix inversion can be used. To find out which are available, see the docstring of the .inv() method. Examples ======== >>> from sympy.matrices import Matrix, ones >>> A = Matrix([1, 2, 3]) >>> B = Matrix([2, 3, 4]) >>> S = Matrix(A.row_join(B)) >>> S Matrix([ [1, 2], [2, 3], [3, 4]]) If each line of S represent coefficients of Ax + By and x and y are [2, 3] then S*xy is: >>> r = S*Matrix([2, 3]); r Matrix([ [ 8], [13], [18]]) But let's add 1 to the middle value and then solve for the least-squares value of xy: >>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy Matrix([ [ 5/3], [10/3]]) The error is given by S*xy - r: >>> S*xy - r Matrix([ [1/3], [1/3], [1/3]]) >>> _.norm().n(2) 0.58 If a different xy is used, the norm will be higher: >>> xy += ones(2, 1)/10 >>> (S*xy - r).norm().n(2) 1.5 """ if method == 'CH': return self.cholesky_solve(rhs) t = self.T return (t * self).inv(method=method) * t * rhs
[docs] def solve(self, rhs, method='GE'): """Return solution to self*soln = rhs using given inversion method. For a list of possible inversion methods, see the .inv() docstring. """ if not self.is_square: if self.rows < self.cols: raise ValueError('Under-determined system. ' 'Try M.gauss_jordan_solve(rhs)') elif self.rows > self.cols: raise ValueError('For over-determined system, M, having ' 'more rows than columns, try M.solve_least_squares(rhs).') else: return self.inv(method=method) * rhs
[docs] def table(self, printer, rowstart='[', rowend=']', rowsep='\n', colsep=', ', align='right'): r""" String form of Matrix as a table. ``printer`` is the printer to use for on the elements (generally something like StrPrinter()) ``rowstart`` is the string used to start each row (by default '['). ``rowend`` is the string used to end each row (by default ']'). ``rowsep`` is the string used to separate rows (by default a newline). ``colsep`` is the string used to separate columns (by default ', '). ``align`` defines how the elements are aligned. Must be one of 'left', 'right', or 'center'. You can also use '<', '>', and '^' to mean the same thing, respectively. This is used by the string printer for Matrix. Examples ======== >>> from sympy import Matrix >>> from sympy.printing.str import StrPrinter >>> M = Matrix([[1, 2], [-33, 4]]) >>> printer = StrPrinter() >>> M.table(printer) '[ 1, 2]\n[-33, 4]' >>> print(M.table(printer)) [ 1, 2] [-33, 4] >>> print(M.table(printer, rowsep=',\n')) [ 1, 2], [-33, 4] >>> print('[%s]' % M.table(printer, rowsep=',\n')) [[ 1, 2], [-33, 4]] >>> print(M.table(printer, colsep=' ')) [ 1 2] [-33 4] >>> print(M.table(printer, align='center')) [ 1 , 2] [-33, 4] >>> print(M.table(printer, rowstart='{', rowend='}')) { 1, 2} {-33, 4} """ # Handle zero dimensions: if self.rows == 0 or self.cols == 0: return '[]' # Build table of string representations of the elements res = [] # Track per-column max lengths for pretty alignment maxlen = [0] * self.cols for i in range(self.rows): res.append([]) for j in range(self.cols): s = printer._print(self[i, j]) res[-1].append(s) maxlen[j] = max(len(s), maxlen[j]) # Patch strings together align = { 'left': 'ljust', 'right': 'rjust', 'center': 'center', '<': 'ljust', '>': 'rjust', '^': 'center', }[align] for i, row in enumerate(res): for j, elem in enumerate(row): row[j] = getattr(elem, align)(maxlen[j]) res[i] = rowstart + colsep.join(row) + rowend return rowsep.join(res)
[docs] def upper_triangular_solve(self, rhs): """Solves Ax = B, where A is an upper triangular matrix. See Also ======== lower_triangular_solve gauss_jordan_solve cholesky_solve diagonal_solve LDLsolve LUsolve QRsolve pinv_solve """ if not self.is_square: raise NonSquareMatrixError("Matrix must be square.") if rhs.rows != self.rows: raise TypeError("Matrix size mismatch.") if not self.is_upper: raise TypeError("Matrix is not upper triangular.") return self._upper_triangular_solve(rhs)
[docs] def vech(self, diagonal=True, check_symmetry=True): """Return the unique elements of a symmetric Matrix as a one column matrix by stacking the elements in the lower triangle. Arguments: diagonal -- include the diagonal cells of self or not check_symmetry -- checks symmetry of self but not completely reliably Examples ======== >>> from sympy import Matrix >>> m=Matrix([[1, 2], [2, 3]]) >>> m Matrix([ [1, 2], [2, 3]]) >>> m.vech() Matrix([ [1], [2], [3]]) >>> m.vech(diagonal=False) Matrix([[2]]) See Also ======== vec """ from sympy.matrices import zeros c = self.cols if c != self.rows: raise ShapeError("Matrix must be square") if check_symmetry: self.simplify() if self != self.transpose(): raise ValueError( "Matrix appears to be asymmetric; consider check_symmetry=False") count = 0 if diagonal: v = zeros(c * (c + 1) // 2, 1) for j in range(c): for i in range(j, c): v[count] = self[i, j] count += 1 else: v = zeros(c * (c - 1) // 2, 1) for j in range(c): for i in range(j + 1, c): v[count] = self[i, j] count += 1 return v
[docs]def classof(A, B): """ Get the type of the result when combining matrices of different types. Currently the strategy is that immutability is contagious. Examples ======== >>> from sympy import Matrix, ImmutableMatrix >>> from sympy.matrices.matrices import classof >>> M = Matrix([[1, 2], [3, 4]]) # a Mutable Matrix >>> IM = ImmutableMatrix([[1, 2], [3, 4]]) >>> classof(M, IM) <class 'sympy.matrices.immutable.ImmutableDenseMatrix'> """ try: if A._class_priority > B._class_priority: return A.__class__ else: return B.__class__ except Exception: pass try: import numpy if isinstance(A, numpy.ndarray): return B.__class__ if isinstance(B, numpy.ndarray): return A.__class__ except Exception: pass raise TypeError("Incompatible classes %s, %s" % (A.__class__, B.__class__))
[docs]def a2idx(j, n=None): """Return integer after making positive and validating against n.""" if type(j) is not int: try: j = j.__index__() except AttributeError: raise IndexError("Invalid index a[%r]" % (j,)) if n is not None: if j < 0: j += n if not (j >= 0 and j < n): raise IndexError("Index out of range: a[%s]" % j) return int(j)
def _find_reasonable_pivot(col, iszerofunc=_iszero, simpfunc=_simplify): """ Find the lowest index of an item in `col` that is suitable for a pivot. If `col` consists only of Floats, the pivot with the largest norm is returned. Otherwise, the first element where `iszerofunc` returns False is used. If `iszerofunc` doesn't return false, items are simplified and retested until a suitable pivot is found. Returns a 4-tuple (pivot_offset, pivot_val, assumed_nonzero, newly_determined) where pivot_offset is the index of the pivot, pivot_val is the (possibly simplified) value of the pivot, assumed_nonzero is True if an assumption that the pivot was non-zero was made without being proved, and newly_determined are elements that were simplified during the process of pivot finding.""" newly_determined = [] col = list(col) # a column that contains a mix of floats and integers # but at least one float is considered a numerical # column, and so we do partial pivoting if all(isinstance(x, (Float, Integer)) for x in col) and any( isinstance(x, Float) for x in col): col_abs = [abs(x) for x in col] max_value = max(col_abs) if iszerofunc(max_value): # just because iszerofunc returned True, doesn't # mean the value is numerically zero. Make sure # to replace all entries with numerical zeros if max_value != 0: newly_determined = [(i, 0) for i, x in enumerate(col) if x != 0] return (None, None, False, newly_determined) index = col_abs.index(max_value) return (index, col[index], False, newly_determined) # PASS 1 (iszerofunc directly) possible_zeros = [] for i, x in enumerate(col): is_zero = iszerofunc(x) # is someone wrote a custom iszerofunc, it may return # BooleanFalse or BooleanTrue instead of True or False, # so use == for comparison instead of `is` if is_zero == False: # we found something that is definitely not zero return (i, x, False, newly_determined) possible_zeros.append(is_zero) # by this point, we've found no certain non-zeros if all(possible_zeros): # if everything is definitely zero, we have # no pivot return (None, None, False, newly_determined) # PASS 2 (iszerofunc after simplify) # we haven't found any for-sure non-zeros, so # go through the elements iszerofunc couldn't # make a determination about and opportunistically # simplify to see if we find something for i, x in enumerate(col): if possible_zeros[i] is not None: continue simped = simpfunc(x) is_zero = iszerofunc(simped) if is_zero == True or is_zero == False: newly_determined.append((i, simped)) if is_zero == False: return (i, simped, False, newly_determined) possible_zeros[i] = is_zero # after simplifying, some things that were recognized # as zeros might be zeros if all(possible_zeros): # if everything is definitely zero, we have # no pivot return (None, None, False, newly_determined) # PASS 3 (.equals(0)) # some expressions fail to simplify to zero, but # `.equals(0)` evaluates to True. As a last-ditch # attempt, apply `.equals` to these expressions for i, x in enumerate(col): if possible_zeros[i] is not None: continue if x.equals(S.Zero): # `.iszero` may return False with # an implicit assumption (e.g., `x.equals(0)` # when `x` is a symbol), so only treat it # as proved when `.equals(0)` returns True possible_zeros[i] = True newly_determined.append((i, S.Zero)) if all(possible_zeros): return (None, None, False, newly_determined) # at this point there is nothing that could definitely # be a pivot. To maintain compatibility with existing # behavior, we'll assume that an illdetermined thing is # non-zero. We should probably raise a warning in this case i = possible_zeros.index(None) return (i, col[i], True, newly_determined) def _find_reasonable_pivot_naive(col, iszerofunc=_iszero, simpfunc=None): """ Helper that computes the pivot value and location from a sequence of contiguous matrix column elements. As a side effect of the pivot search, this function may simplify some of the elements of the input column. A list of these simplified entries and their indices are also returned. This function mimics the behavior of _find_reasonable_pivot(), but does less work trying to determine if an indeterminate candidate pivot simplifies to zero. This more naive approach can be much faster, with the trade-off that it may erroneously return a pivot that is zero. `col` is a sequence of contiguous column entries to be searched for a suitable pivot. `iszerofunc` is a callable that returns a Boolean that indicates if its input is zero, or None if no such determination can be made. `simpfunc` is a callable that simplifies its input. It must return its input if it does not simplify its input. Passing in `simpfunc=None` indicates that the pivot search should not attempt to simplify any candidate pivots. Returns a 4-tuple: (pivot_offset, pivot_val, assumed_nonzero, newly_determined) `pivot_offset` is the sequence index of the pivot. `pivot_val` is the value of the pivot. pivot_val and col[pivot_index] are equivalent, but will be different when col[pivot_index] was simplified during the pivot search. `assumed_nonzero` is a boolean indicating if the pivot cannot be guaranteed to be zero. If assumed_nonzero is true, then the pivot may or may not be non-zero. If assumed_nonzero is false, then the pivot is non-zero. `newly_determined` is a list of index-value pairs of pivot candidates that were simplified during the pivot search. """ # indeterminates holds the index-value pairs of each pivot candidate # that is neither zero or non-zero, as determined by iszerofunc(). # If iszerofunc() indicates that a candidate pivot is guaranteed # non-zero, or that every candidate pivot is zero then the contents # of indeterminates are unused. # Otherwise, the only viable candidate pivots are symbolic. # In this case, indeterminates will have at least one entry, # and all but the first entry are ignored when simpfunc is None. indeterminates = [] for i, col_val in enumerate(col): col_val_is_zero = iszerofunc(col_val) if col_val_is_zero == False: # This pivot candidate is non-zero. return i, col_val, False, [] elif col_val_is_zero is None: # The candidate pivot's comparison with zero # is indeterminate. indeterminates.append((i, col_val)) if len(indeterminates) == 0: # All candidate pivots are guaranteed to be zero, i.e. there is # no pivot. return None, None, False, [] if simpfunc is None: # Caller did not pass in a simplification function that might # determine if an indeterminate pivot candidate is guaranteed # to be nonzero, so assume the first indeterminate candidate # is non-zero. return indeterminates[0][0], indeterminates[0][1], True, [] # newly_determined holds index-value pairs of candidate pivots # that were simplified during the search for a non-zero pivot. newly_determined = [] for i, col_val in indeterminates: tmp_col_val = simpfunc(col_val) if id(col_val) != id(tmp_col_val): # simpfunc() simplified this candidate pivot. newly_determined.append((i, tmp_col_val)) if iszerofunc(tmp_col_val) == False: # Candidate pivot simplified to a guaranteed non-zero value. return i, tmp_col_val, False, newly_determined return indeterminates[0][0], indeterminates[0][1], True, newly_determined class _MinimalMatrix(object): """Class providing the minimum functionality for a matrix-like object and implementing every method required for a `MatrixRequired`. This class does not have everything needed to become a full-fledged sympy object, but it will satisfy the requirements of anything inheriting from `MatrixRequired`. If you wish to make a specialized matrix type, make sure to implement these methods and properties with the exception of `__init__` and `__repr__` which are included for convenience.""" is_MatrixLike = True _sympify = staticmethod(sympify) _class_priority = 3 is_Matrix = True is_MatrixExpr = False @classmethod def _new(cls, *args, **kwargs): return cls(*args, **kwargs) def __init__(self, rows, cols=None, mat=None): if isinstance(mat, FunctionType): # if we passed in a function, use that to populate the indices mat = list(mat(i, j) for i in range(rows) for j in range(cols)) try: if cols is None and mat is None: mat = rows rows, cols = mat.shape except AttributeError: pass try: # if we passed in a list of lists, flatten it and set the size if cols is None and mat is None: mat = rows cols = len(mat[0]) rows = len(mat) mat = [x for l in mat for x in l] except (IndexError, TypeError): pass self.mat = tuple(self._sympify(x) for x in mat) self.rows, self.cols = rows, cols if self.rows is None or self.cols is None: raise NotImplementedError("Cannot initialize matrix with given parameters") def __getitem__(self, key): def _normalize_slices(row_slice, col_slice): """Ensure that row_slice and col_slice don't have `None` in their arguments. Any integers are converted to slices of length 1""" if not isinstance(row_slice, slice): row_slice = slice(row_slice, row_slice + 1, None) row_slice = slice(*row_slice.indices(self.rows)) if not isinstance(col_slice, slice): col_slice = slice(col_slice, col_slice + 1, None) col_slice = slice(*col_slice.indices(self.cols)) return (row_slice, col_slice) def _coord_to_index(i, j): """Return the index in _mat corresponding to the (i,j) position in the matrix. """ return i * self.cols + j if isinstance(key, tuple): i, j = key if isinstance(i, slice) or isinstance(j, slice): # if the coordinates are not slices, make them so # and expand the slices so they don't contain `None` i, j = _normalize_slices(i, j) rowsList, colsList = list(range(self.rows))[i], \ list(range(self.cols))[j] indices = (i * self.cols + j for i in rowsList for j in colsList) return self._new(len(rowsList), len(colsList), list(self.mat[i] for i in indices)) # if the key is a tuple of ints, change # it to an array index key = _coord_to_index(i, j) return self.mat[key] def __eq__(self, other): return self.shape == other.shape and list(self) == list(other) def __len__(self): return self.rows*self.cols def __repr__(self): return "_MinimalMatrix({}, {}, {})".format(self.rows, self.cols, self.mat) @property def shape(self): return (self.rows, self.cols) class _MatrixWrapper(object): """Wrapper class providing the minimum functionality for a matrix-like object: .rows, .cols, .shape, indexability, and iterability. CommonMatrix math operations should work on matrix-like objects. For example, wrapping a numpy matrix in a MatrixWrapper allows it to be passed to CommonMatrix. """ is_MatrixLike = True def __init__(self, mat, shape=None): self.mat = mat self.rows, self.cols = mat.shape if shape is None else shape def __getattr__(self, attr): """Most attribute access is passed straight through to the stored matrix""" return getattr(self.mat, attr) def __getitem__(self, key): return self.mat.__getitem__(key) def _matrixify(mat): """If `mat` is a Matrix or is matrix-like, return a Matrix or MatrixWrapper object. Otherwise `mat` is passed through without modification.""" if getattr(mat, 'is_Matrix', False): return mat if hasattr(mat, 'shape'): if len(mat.shape) == 2: return _MatrixWrapper(mat) return mat