Commutation matrix

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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):

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:

vec(A) = [ A1,1, ..., Am,1, A1,2, ..., Am,2, ..., A1,n, ..., Am,n ]T

where A = [Ai,j].

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,

K(r,m)(A \otimes B)K(n,q) = B \otimes A.

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

K(r,m) = \sum_{i=1}^{r}\sum_{j=1}^{m}er,iem,jT\otimesem,jer,iT.


Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley.