Dense Matrices

sympy.matrices.dense.Matrix

alias of MutableDenseMatrix

class sympy.matrices.dense.DenseMatrix

Matrix implementation based on DomainMatrix as the internal representation

LDLdecomposition(hermitian=True)

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 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

See also

sympy.matrices.dense.DenseMatrix.cholesky, sympy.matrices.matrixbase.MatrixBase.LUdecomposition, QRdecomposition

as_immutable()

Returns an Immutable version of this Matrix

as_mutable()

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)

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 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

See also

sympy.matrices.dense.DenseMatrix.LDLdecomposition, sympy.matrices.matrixbase.MatrixBase.LUdecomposition, QRdecomposition

lower_triangular_solve(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, cramer_solve

upper_triangular_solve(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, cramer_solve

class sympy.matrices.dense.MutableDenseMatrix(*args, **kwargs)

simplify(**kwargs)

Applies simplify to the elements of a matrix in place.

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

See also

sympy.simplify.simplify.simplify

class sympy.matrices.immutable.ImmutableDenseMatrix(*args, **kwargs)

Create an immutable version of a matrix.

Examples

>>> from sympy import eye, 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