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) = T
where A = .
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 B)K(n,q) = B A.
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