# 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[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
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[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
class sympy.matrices.expressions.MatAdd[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
class sympy.matrices.expressions.MatMul[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
class sympy.matrices.expressions.MatPow[source]

Attributes

 is_Identity
class sympy.matrices.expressions.Inverse[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
class sympy.matrices.expressions.Transpose[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
class sympy.matrices.expressions.Trace[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
class sympy.matrices.expressions.FunctionMatrix[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[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[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[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]])

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[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]])