Commutant

Commutant
Computes the commutant of a set of matrices
Other toolboxes required	opt_args PartialTranspose PermuteSystems spnull Swap

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.