Matrix Expressions

The Matrix expression module allows users to write down statements like

>>> from sympy import MatrixSymbol, Matrix
>>> X = MatrixSymbol('X', 3, 3)
>>> Y = MatrixSymbol('Y', 3, 3)
>>> (X.T*X).I*Y
X^-1*X.T^-1*Y

>>> Matrix(X)
Matrix([
[X[0, 0], X[0, 1], X[0, 2]],
[X[1, 0], X[1, 1], X[1, 2]],
[X[2, 0], X[2, 1], X[2, 2]]])

>>> (X*Y)[1, 2]
X[1, 0]*Y[0, 2] + X[1, 1]*Y[1, 2] + X[1, 2]*Y[2, 2]

where X and Y are MatrixSymbol’s rather than scalar symbols.

Matrix Expressions Core Reference

class sympy.matrices.expressions.MatrixExpr(*args, **kwargs)[source]

Superclass for Matrix Expressions

MatrixExprs represent abstract matrices, linear transformations represented within a particular basis.

Examples

>>> from sympy import MatrixSymbol
>>> A = MatrixSymbol('A', 3, 3)
>>> y = MatrixSymbol('y', 3, 1)
>>> x = (A.T*A).I * A * y

Attributes

is_Identity
property T

Matrix transposition.

as_coeff_Mul(rational=False)[source]

Efficiently extract the coefficient of a product.

as_explicit()[source]

Returns a dense Matrix with elements represented explicitly

Returns an object of type ImmutableDenseMatrix.

See also

as_mutable

returns mutable Matrix type

Examples

>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
as_mutable()[source]

Returns a dense, mutable matrix with elements represented explicitly

See also

as_explicit

returns ImmutableDenseMatrix

Examples

>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.shape
(3, 3)
>>> I.as_mutable()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
equals(other)[source]

Test elementwise equality between matrices, potentially of different types

>>> from sympy import Identity, eye
>>> Identity(3).equals(eye(3))
True
class sympy.matrices.expressions.MatrixSymbol(name, n, m)[source]

Symbolic representation of a Matrix object

Creates a SymPy Symbol to represent a Matrix. This matrix has a shape and can be included in Matrix Expressions

>>> from sympy import MatrixSymbol, Identity
>>> A = MatrixSymbol('A', 3, 4) # A 3 by 4 Matrix
>>> B = MatrixSymbol('B', 4, 3) # A 4 by 3 Matrix
>>> A.shape
(3, 4)
>>> 2*A*B + Identity(3)
I + 2*A*B

Attributes

is_Identity
doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
property free_symbols

Return from the atoms of self those which are free symbols.

For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method.

Any other method that uses bound variables should implement a free_symbols method.

class sympy.matrices.expressions.MatAdd(*args, **kwargs)[source]

A Sum of Matrix Expressions

MatAdd inherits from and operates like SymPy Add

>>> from sympy import MatAdd, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> C = MatrixSymbol('C', 5, 5)
>>> MatAdd(A, B, C)
A + B + C

Attributes

is_Identity
doit(**kwargs)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.matrices.expressions.MatMul(*args, **kwargs)[source]

A product of matrix expressions

Examples

>>> from sympy import MatMul, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 4)
>>> B = MatrixSymbol('B', 4, 3)
>>> C = MatrixSymbol('C', 3, 6)
>>> MatMul(A, B, C)
A*B*C

Attributes

is_Identity
doit(**kwargs)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.matrices.expressions.MatPow(base, exp)[source]

Attributes

is_Identity
doit(**kwargs)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.matrices.expressions.Inverse(mat)[source]

The multiplicative inverse of a matrix expression

This is a symbolic object that simply stores its argument without evaluating it. To actually compute the inverse, use the .inverse() method of matrices.

Examples

>>> from sympy import MatrixSymbol, Inverse
>>> A = MatrixSymbol('A', 3, 3)
>>> B = MatrixSymbol('B', 3, 3)
>>> Inverse(A)
A^-1
>>> A.inverse() == Inverse(A)
True
>>> (A*B).inverse()
B^-1*A^-1
>>> Inverse(A*B)
(A*B)^-1

Attributes

is_Identity
doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.matrices.expressions.Transpose(*args, **kwargs)[source]

The transpose of a matrix expression.

This is a symbolic object that simply stores its argument without evaluating it. To actually compute the transpose, use the transpose() function, or the .T attribute of matrices.

Examples

>>> from sympy.matrices import MatrixSymbol, Transpose
>>> from sympy.functions import transpose
>>> A = MatrixSymbol('A', 3, 5)
>>> B = MatrixSymbol('B', 5, 3)
>>> Transpose(A)
A.T
>>> A.T == transpose(A) == Transpose(A)
True
>>> Transpose(A*B)
(A*B).T
>>> transpose(A*B)
B.T*A.T

Attributes

is_Identity
doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.matrices.expressions.Trace(mat)[source]

Matrix Trace

Represents the trace of a matrix expression.

>>> from sympy import MatrixSymbol, Trace, eye
>>> A = MatrixSymbol('A', 3, 3)
>>> Trace(A)
Trace(A)
See Also:

trace

doit(**kwargs)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.matrices.expressions.FunctionMatrix(rows, cols, lamda)[source]

Represents a Matrix using a function (Lambda)

This class is an alternative to SparseMatrix

>>> from sympy import FunctionMatrix, symbols, Lambda, MatMul, Matrix
>>> i, j = symbols('i,j')
>>> X = FunctionMatrix(3, 3, Lambda((i, j), i + j))
>>> Matrix(X)
Matrix([
[0, 1, 2],
[1, 2, 3],
[2, 3, 4]])

>>> Y = FunctionMatrix(1000, 1000, Lambda((i, j), i + j))

>>> isinstance(Y*Y, MatMul) # this is an expression object
True

>>> (Y**2)[10,10] # So this is evaluated lazily
342923500

Attributes

is_Identity
class sympy.matrices.expressions.Identity(n)[source]

The Matrix Identity I - multiplicative identity

>>> from sympy.matrices import Identity, MatrixSymbol
>>> A = MatrixSymbol('A', 3, 5)
>>> I = Identity(3)
>>> I*A
A
class sympy.matrices.expressions.ZeroMatrix(m, n)[source]

The Matrix Zero 0 - additive identity

>>> from sympy import MatrixSymbol, ZeroMatrix
>>> A = MatrixSymbol('A', 3, 5)
>>> Z = ZeroMatrix(3, 5)
>>> A+Z
A
>>> Z*A.T
0

Attributes

is_Identity

Block Matrices

Block matrices allow you to construct larger matrices out of smaller sub-blocks. They can work with MatrixExpr or ImmutableMatrix objects.

class sympy.matrices.expressions.blockmatrix.BlockMatrix(*args)[source]

A BlockMatrix is a Matrix composed of other smaller, submatrices

The submatrices are stored in a SymPy Matrix object but accessed as part of a Matrix Expression

>>> from sympy import (MatrixSymbol, BlockMatrix, symbols,
...     Identity, ZeroMatrix, block_collapse)
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])

>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])

>>> print(block_collapse(C*B))
Matrix([[X, Z*Y + Z]])
equals(other)[source]

Test elementwise equality between matrices, potentially of different types

>>> from sympy import Identity, eye
>>> Identity(3).equals(eye(3))
True
transpose()[source]

Return transpose of matrix.

Examples

>>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix
>>> from sympy.abc import l, m, n
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> B.transpose()
Matrix([
[X.T,  0],
[Z.T, Y.T]])
>>> _.transpose()
Matrix([
[X, Z],
[0, Y]])
class sympy.matrices.expressions.blockmatrix.BlockDiagMatrix(*mats)[source]

A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal

>>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols, Identity
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> BlockDiagMatrix(X, Y)
Matrix([
[X, 0],
[0, Y]])
sympy.matrices.expressions.blockmatrix.block_collapse(expr)[source]

Evaluates a block matrix expression

>>> from sympy import MatrixSymbol, BlockMatrix, symbols,                           Identity, Matrix, ZeroMatrix, block_collapse
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])

>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])

>>> print(block_collapse(C*B))
Matrix([[X, Z*Y + Z]])