# 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(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 ${\displaystyle \otimes }$ B)K(n,q) = B ${\displaystyle \otimes }$ 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) = ${\displaystyle \sum _{i=1}^{r}}$${\displaystyle \sum _{j=1}^{m}}$(er,iem,jT)${\displaystyle \otimes }$(em,jer,iT).

## Example

Let M be a 2x2 square matrix.

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

Then we have

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

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

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

## References

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