Dense Matrices

Matrix Class Reference

sympy.matrices.dense.Matrix[source]

alias of sympy.matrices.dense.MutableDenseMatrix

class sympy.matrices.dense.DenseMatrix[source]
LDLdecomposition(hermitian=True)[source]

Returns the LDL Decomposition (L, D) of matrix A, such that L * D * L.H == A if hermitian flag is True, or L * D * L.T == A if hermitian is False. This method eliminates the use of square root. Further this ensures that all the diagonal entries of L are 1. A must be a Hermitian positive-definite matrix if hermitian is True, or a symmetric matrix otherwise.

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

The matrix can have complex entries:

>>> from sympy import I
>>> A = Matrix(((9, 3*I), (-3*I, 5)))
>>> L, D = A.LDLdecomposition()
>>> L
Matrix([
[   1, 0],
[-I/3, 1]])
>>> D
Matrix([
[9, 0],
[0, 4]])
>>> L*D*L.H == A
True
as_immutable()[source]

Returns an Immutable version of this Matrix

as_mutable()[source]

Returns a mutable version of this matrix

Examples

>>> from sympy import ImmutableMatrix
>>> X = ImmutableMatrix([[1, 2], [3, 4]])
>>> Y = X.as_mutable()
>>> Y[1, 1] = 5 # Can set values in Y
>>> Y
Matrix([
[1, 2],
[3, 5]])
cholesky(hermitian=True)[source]

Returns the Cholesky-type decomposition L of a matrix A such that L * L.H == A if hermitian flag is True, or L * L.T == A if hermitian is False.

A must be a Hermitian positive-definite matrix if hermitian is True, or a symmetric matrix if it is False.

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]])

The matrix can have complex entries:

>>> from sympy import I
>>> A = Matrix(((9, 3*I), (-3*I, 5)))
>>> A.cholesky()
Matrix([
[ 3, 0],
[-I, 2]])
>>> A.cholesky() * A.cholesky().H
Matrix([
[   9, 3*I],
[-3*I,   5]])

Non-hermitian Cholesky-type decomposition may be useful when the matrix is not positive-definite.

>>> A = Matrix([[1, 2], [2, 1]])
>>> L = A.cholesky(hermitian=False)
>>> L
Matrix([
[1,         0],
[2, sqrt(3)*I]])
>>> L*L.T == A
True
equals(other, failing_expression=False)[source]

Applies equals to corresponding elements of the matrices, trying to prove that the elements are equivalent, returning True if they are, False if any pair is not, and None (or the first failing expression if failing_expression is True) if it cannot be decided if the expressions are equivalent or not. This is, in general, an expensive operation.

Examples

>>> from sympy.matrices import Matrix
>>> from sympy.abc import x
>>> from sympy import cos
>>> A = Matrix([x*(x - 1), 0])
>>> B = Matrix([x**2 - x, 0])
>>> A == B
False
>>> A.simplify() == B.simplify()
True
>>> A.equals(B)
True
>>> A.equals(2)
False
lower_triangular_solve(rhs)[source]

Solves Ax = B, where A is a lower triangular matrix.

upper_triangular_solve(rhs)[source]

Solves Ax = B, where A is an upper triangular matrix.

class sympy.matrices.dense.MutableDenseMatrix[source]
col_del(i)[source]

Delete the given column.

Examples

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.col_del(1)
>>> M
Matrix([
[1, 0],
[0, 0],
[0, 1]])

See also

col, row_del

col_op(j, f)[source]

In-place operation on col j using two-arg functor whose args are interpreted as (self[i, j], i).

Examples

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M
Matrix([
[1, 2, 0],
[0, 1, 0],
[0, 0, 1]])

See also

col, row_op

col_swap(i, j)[source]

Swap the two given columns of the matrix in-place.

Examples

>>> from sympy.matrices import Matrix
>>> M = Matrix([[1, 0], [1, 0]])
>>> M
Matrix([
[1, 0],
[1, 0]])
>>> M.col_swap(0, 1)
>>> M
Matrix([
[0, 1],
[0, 1]])

See also

col, row_swap

copyin_list(key, value)[source]

Copy in elements from a list.

Parameters

key : slice

The section of this matrix to replace.

value : iterable

The iterable to copy values from.

Examples

>>> from sympy.matrices import eye
>>> I = eye(3)
>>> I[:2, 0] = [1, 2] # col
>>> I
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
>>> I[1, :2] = [[3, 4]]
>>> I
Matrix([
[1, 0, 0],
[3, 4, 0],
[0, 0, 1]])

See also

copyin_matrix

copyin_matrix(key, value)[source]

Copy in values from a matrix into the given bounds.

Parameters

key : slice

The section of this matrix to replace.

value : Matrix

The matrix to copy values from.

Examples

>>> from sympy.matrices import Matrix, eye
>>> M = Matrix([[0, 1], [2, 3], [4, 5]])
>>> I = eye(3)
>>> I[:3, :2] = M
>>> I
Matrix([
[0, 1, 0],
[2, 3, 0],
[4, 5, 1]])
>>> I[0, 1] = M
>>> I
Matrix([
[0, 0, 1],
[2, 2, 3],
[4, 4, 5]])

See also

copyin_list

fill(value)[source]

Fill the matrix with the scalar value.

See also

zeros, ones

row_del(i)[source]

Delete the given row.

Examples

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.row_del(1)
>>> M
Matrix([
[1, 0, 0],
[0, 0, 1]])

See also

row, col_del

row_op(i, f)[source]

In-place operation on row i using two-arg functor whose args are interpreted as (self[i, j], j).

Examples

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])

See also

row, zip_row_op, col_op

row_swap(i, j)[source]

Swap the two given rows of the matrix in-place.

Examples

>>> from sympy.matrices import Matrix
>>> M = Matrix([[0, 1], [1, 0]])
>>> M
Matrix([
[0, 1],
[1, 0]])
>>> M.row_swap(0, 1)
>>> M
Matrix([
[1, 0],
[0, 1]])

See also

row, col_swap

simplify(**kwargs)[source]

Applies simplify to the elements of a matrix in place.

This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))

zip_row_op(i, k, f)[source]

In-place operation on row i using two-arg functor whose args are interpreted as (self[i, j], self[k, j]).

Examples

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])

See also

row, row_op, col_op

ImmutableMatrix Class Reference

class sympy.matrices.immutable.ImmutableDenseMatrix[source]

Create an immutable version of a matrix.

Examples

>>> from sympy import eye
>>> from sympy.matrices import ImmutableMatrix
>>> ImmutableMatrix(eye(3))
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> _[0, 0] = 42
Traceback (most recent call last):
...
TypeError: Cannot set values of ImmutableDenseMatrix
is_diagonalizable(reals_only=False, **kwargs)[source]

Returns True if a matrix is diagonalizable.

Parameters

reals_only : bool, optional

If True, it tests whether the matrix can be diagonalized to contain only real numbers on the diagonal.

If False, it tests whether the matrix can be diagonalized at all, even with numbers that may not be real.

Examples

Example of a diagonalizable matrix:

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

Example of a non-diagonalizable matrix:

>>> M = Matrix([[0, 1], [0, 0]])
>>> M.is_diagonalizable()
False

Example of a matrix that is diagonalized in terms of non-real entries:

>>> M = Matrix([[0, 1], [-1, 0]])
>>> M.is_diagonalizable(reals_only=False)
True
>>> M.is_diagonalizable(reals_only=True)
False