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 ${\displaystyle \otimes }$ B)K^(n,q) = B ${\displaystyle \otimes }$ A.

It is much used in developing the higher order statistics of Wishart covariance matrices.[1]

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) = ${\displaystyle \sum _{i=1}^{r}}$${\displaystyle \sum _{j=1}^{m}}$(e_r,ie_m,j^T)${\displaystyle \otimes }$(e_m,je_r,i^T).

Example

Let M be a 2x2 square matrix.

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

Then we have

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

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

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

For both square and rectangular matrices of ${\displaystyle M{\text{ rows and }}N}$ columns, the commutation matrix can be generated by this generic pseudo-code, which is similar to an article at StackExchange.com[2] and demonstrably gives the correct result though is presented without proof.

 for i = 1 to M
     for j = 1 to N
        K(i + M*(j - 1), j + N*(i - 1)) = 1
     end
  end

Thus the following ${\displaystyle 3\times 2{\text{ matrix }}A}$ has two possible vectorizations as follows:

${\displaystyle A={\begin{bmatrix}1&4\\2&5\\3&6\\\end{bmatrix}},\;\;\;\;V1=\mathbf {vec(A)} ={\begin{bmatrix}1\\2\\3\\4\\5\\6\\\end{bmatrix}},\;\;\;\;\;V2=\mathbf {vec(A^{T})} ={\begin{bmatrix}1\\4\\2\\5\\3\\6\\\end{bmatrix}}}$

and the code above yields

${\displaystyle K={\begin{bmatrix}1&\cdot &\cdot &\cdot &\cdot &\cdot &\\\cdot &\cdot &1&\cdot &\cdot &\cdot &\\\cdot &\cdot &\cdot &\cdot &1&\cdot &\\\cdot &1&\cdot &\cdot &\cdot &\cdot &\\\cdot &\cdot &\cdot &1&\cdot &\cdot &\\\cdot &\cdot &\cdot &\cdot &\cdot &1&\\\end{bmatrix}}}$

giving the expected results

${\displaystyle K^{T}K=KK^{T}=\mathbf {I} _{6\times 6}}$

${\displaystyle K^{T}\times V1=V2}$

${\displaystyle K\times V2=V1}$

References

1. von Rosen, Dietrich (1988). "Moments for the Inverted Wishart Distribution". Scand J Statistics. 15: 97–109.
2. "Kronecker product and the commutation matrix". Stack Exchange. 2013. |first= missing |last= (help)

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