About: Commutation matrix

In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): K(m,n) vec(A) = vec(AT) . Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another: In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator