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 mn × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): 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 mn × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another: where A = . The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. Replacing A with AT in the definition of the commutation matrix shows that K(m,n) = (K(n,m))T. Therefore in the special case of m = n the commutation matrix is an involution and symmetric. The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product: for every m × n matrix A and every r × q matrix B, An explicit form for the commutation matrix is as follows: if er,j denotes the j-th canonical vector of dimension r (i.e. the vector with 1 in the j-th coordinate and 0 elsewhere) then Let M be a 2x2 square matrix. M = [ a b c d ] {displaystyle mathbf {M} ={egin{bmatrix}a&b\c&d\end{bmatrix}}}