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(A^T):

K^(m,n) vec(A) = vec(A^T) .

Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another:

vec(A) = [ A_1,1, ..., A_m,1, A_1,2, ..., A_m,2, ..., A_1,n, ..., A_m,n ]^T

where A = [A_i,j].

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.

References

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