toqito.matrix_props.commutant

Module for computing the commutant of a set of matrices.

Module Contents

toqito.matrix_props.commutant.commutant(A)

Compute an orthonormal basis for the commutant algebra [@PlanetMathCommutant].

Given a matrix (A) or a set of matrices (mathcal{A} = {A_1, A_2, dots}), this function determines an orthonormal basis (with respect to the Hilbert-Schmidt inner product) for the algebra of matrices that commute with every matrix in (mathcal{A}).

The commutant of a single matrix (A in mathbb{C}^{n times n}) consists of all matrices (X in mathbb{C}^{n times n}) satisfying:

[

A X = X A.

]

More generally, for a set of matrices (mathcal{A} = {A_1, A_2, dots}), the commutant consists of all matrices (X) satisfying:

[

A_i X = X A_i quad forall A_i in mathcal{A}.

]

This condition can be rewritten in vectorized form as:

[

(A_i otimes I - I otimes A_i^T) text{vec}(X) = 0, quad forall A_i in mathcal{A}.

]

where (text{vec}(X)) denotes the column-wise vectorization of (X). The null space of this equation provides a basis for the commutant.

This implementation is based on [@QETLAB_link].

Examples

Consider the following set of matrices:

[

A_1 = begin{pmatrix}

1 & 0 \ 0 & -1

end{pmatrix}, quad

A_2 = begin{pmatrix}

0 & 1 \ 1 & 0

end{pmatrix}

]

The commutant consists of matrices that commute with both (A_1) and (A_2).

```python exec=”1” source=”above” import numpy as np from toqito.matrix_props import commutant

A1 = np.array([[1, 0], [0, -1]]) A2 = np.array([[0, 1], [1, 0]])

basis = commutant([A1, A2])

print(basis) ```

Now, consider a single matrix:

[

A = begin{pmatrix}

1 & 1 \ 0 & 1

end{pmatrix}

]

```python exec=”1” source=”above” import numpy as np from toqito.matrix_props import commutant

A = np.array([[1, 1], [0, 1]])

basis = commutant(A)

for i, basis_ in enumerate(basis):

print(f”basis{ i} :n{basis_} n”)

```

Parameters:

A (numpy.ndarray | list[numpy.ndarray]) – A single matrix of the form np.ndarray or a list of square matrices of the same dimension.

Returns:

A list of matrices forming an orthonormal basis for the commutant.

Return type:

list[numpy.ndarray]