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 nm × 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

\otimes

B)K^(n,q) = B

\otimes

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) =

\sum _{i=1}^{r}

\sum _{j=1}^{m}

(e_r,ie_m,j^T)

\otimes

(e_m,je_r,i^T).

Example

Let M be a 2x2 square matrix.

$\mathbf {M} ={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}$

Then we have

$\operatorname {vec} (\mathbf {M} )={\begin{bmatrix}a\\c\\b\\d\\\end{bmatrix}}$

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

${\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{bmatrix}}\cdot {\begin{bmatrix}a\\c\\b\\d\\\end{bmatrix}}={\begin{bmatrix}a\\b\\c\\d\\\end{bmatrix}}=\operatorname {vec} (\mathbf {M} ^{T})$

References

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

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