# Matrix Expressions¶

The Matrix expression module allows users to write down statements like

>>> X = MatrixSymbol('X', 3, 3)
>>> Y = MatrixSymbol('Y', 3, 3)
>>> (X.T*X).inverse()*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 diofant.matrices.expressions.MatrixExpr[source]

Superclass for Matrix Expressions

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

Examples

>>> A = MatrixSymbol('A', 3, 3)
>>> y = MatrixSymbol('y', 3, 1)
>>> x = (A.T*A).inverse() * A * y

T

Matrix transposition.

as_explicit()[source]

Returns a dense Matrix with elements represented explicitly

Returns an object of type ImmutableMatrix.

Examples

>>> I = Identity(3)
>>> I
I
>>> I.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])


as_mutable()

returns mutable Matrix type

as_mutable()[source]

Returns a dense, mutable matrix with elements represented explicitly

Examples

>>> I = Identity(3)
>>> I
I
>>> I.shape
(3, 3)
>>> I.as_mutable()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])


as_explicit()

returns ImmutableMatrix

equals(other)[source]

Test elementwise equality between matrices, potentially of different types

>>> Identity(3).equals(eye(3))
True

class diofant.matrices.expressions.MatrixSymbol[source]

Symbolic representation of a Matrix object

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

>>> 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

class diofant.matrices.expressions.MatAdd[source]

A Sum of Matrix Expressions

>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> C = MatrixSymbol('C', 5, 5)
A + B + C

class diofant.matrices.expressions.MatMul[source]

A product of matrix expressions

Examples

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

class diofant.matrices.expressions.MatPow[source]
class diofant.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

>>> 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

class diofant.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

>>> 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

class diofant.matrices.expressions.Trace[source]

Matrix Trace

Represents the trace of a matrix expression.

>>> A = MatrixSymbol('A', 3, 3)
>>> Trace(A)
Trace(A)

class diofant.matrices.expressions.FunctionMatrix[source]

Represents a Matrix using a function (Lambda)

This class is an alternative to SparseMatrix

>>> 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

class diofant.matrices.expressions.Identity[source]

The Matrix Identity I - multiplicative identity

>>> A = MatrixSymbol('A', 3, 5)
>>> I = Identity(3)
>>> I*A
A

class diofant.matrices.expressions.ZeroMatrix[source]

The Matrix Zero 0 - additive identity

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


## Block Matrices¶

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

class diofant.matrices.expressions.blockmatrix.BlockMatrix[source]

A BlockMatrix is a Matrix composed of other smaller, submatrices

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

>>> 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]])
>>> B
Matrix([
[X, Z],
[0, Y]])

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

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

transpose()[source]

Return transpose of matrix.

Examples

>>> from diofant.abc import l
>>> 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 diofant.matrices.expressions.blockmatrix.BlockDiagMatrix[source]

A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal

>>> 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]])

diofant.matrices.expressions.blockmatrix.block_collapse(expr)[source]

Evaluates a block matrix expression

>>> 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]])
>>> B
Matrix([
[X, Z],
[0, Y]])

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

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