Commutant

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