# Commutant

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Commutant | |

Computes the commutant of a set of matrices | |

Other toolboxes required | none |
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Function category | Miscellaneous |

` Commutant` is a function that computes an orthonormal basis (in the Hilbert-Schmidt inner product) of the commutant of a matrix or set of matrices.

## Syntax

`C = Commutant(A)`

## Argument descriptions

`A`: Either a matrix, or a cell containing one or more matrices of the same size.

## Examples

### A single matrix

Everything commutes with the identity matrix, which we can verify in the 2-dimensional case as follows:

```
>> C = Commutant(eye(2));
>> celldisp(C) % display the contents of C
C{1} =
1 0
0 0
C{2} =
0 1
0 0
C{3} =
0 0
1 0
C{4} =
0 0
0 1
```

### Bicommutant

If `A` is a cell that contains a basis of a unital, self-adjoint algebra, then its bicommutant spans the same set, by the von Neumann bicommutant theorem. The following code verifies this in the case of the subalgebra $M_2 \oplus M_1 \subseteq M_3$:

```
>> A = {[1 0 0;0 0 0;0 0 0] [0 1 0;0 0 0;0 0 0] [0 0 0;1 0 0;0 0 0] [0 0 0;0 1 0;0 0 0] [0 0 0;0 0 0;0 0 1]};
>> B = Commutant(Commutant(A));
>> celldisp(B)
B{1} =
1 0 0
0 0 0
0 0 0
B{2} =
0 1 0
0 0 0
0 0 0
B{3} =
0 0 0
1 0 0
0 0 0
B{4} =
0 0 0
0 1 0
0 0 0
B{5} =
0 0 0
0 0 0
0 0 1
```

Note that the basis output for the bicommutant in general may be different than the basis provided as input.

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.