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.

1. %%  COMMUTANT    Computes the commutant of a set of matrices

2. %   This function has one required argument:

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

4. %

5. %   C = Commutant(A) is a cell containing an orthonormal basis (in the

6. %   Hilbert-Schmidt inner product) for the algebra of matrices that commute

7. %   with each matrix in the cell A. The elements of C are sparse if and

8. %   only if the elements of A are sparse.

9. %

10. %   URL: http://www.qetlab.com/Commutant

12. %   requires: opt_args.m, PartialTranspose.m, PermuteSystems.m, spnull.m,

13. %             Swap.m

14. %

15. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

16. %   package: QETLAB

17. %   last updated: November 12, 2014

19. function C = Commutant(A)

21. if(iscell(A)) % allow A to just be a matrix instead of a cell if they only want the commutant of one matrix

22.     A = [A{:}].';

23. else

24.     A = A.';

25. end

26. dim = size(A,2);

27. num_ops = length(A)/dim;

29. % a sneaky (and fast) way of constructing the commutant that works by

30. % noting that AX = XA if and only if (kron(A,I) - kron(I,A^T))x = 0, where

31. % x is the vectorization of X

32. C = reshape(spnull( kron(PartialTranspose(A,2,[num_ops,dim;1,dim]),speye(dim)) - Swap(kron(speye(dim),A),[1,2],[dim,num_ops,dim],1) ),dim,[]);

34. % from here on, we're just reshaping the resulting data into matrices of

35. % the proper size and sparsity

36. num_comm_ops = size(C,2)/dim;

37. if(~issparse(A))

38.     C = full(C);

39. end

40. C = mat2cell(PartialTranspose(C,2,[1,dim;num_comm_ops,dim]),dim,dim*ones(1,num_comm_ops));