# Difference between revisions of "Commutant"

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 Other toolboxes required Commutant Computes the commutant of a set of matrices none 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.