Commutation matrix - Alchetron, The Free Social Encyclopedia

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 commutation matrix is a special type of permutation matrix, and is therefore orthogonal. Replacing A with A^T 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 ⊗ B)K^(n,q) = B ⊗ A.

An explicit form for the commutation matrix is as follows: if e_r,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) = ∑ i = 1 r ∑ j = 1 m (e_r,ie_m,j^T) ⊗ (e_m,je_r,i^T).

Example

Let M be a 2x2 square matrix.

M = [ a b c d ]

Then we have

v e c ( M ) = [ a c b d ]

And K^(2,2) is the 4x4 square matrix that will transform vec(M) into vec(M^T)

[ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] ⋅ [ a c b d ] = [ a b c d ] = v e c ( M T )

References

Commutation matrix Wikipedia

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