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:

$\operatorname {vec} (\mathbf {A} )=[\mathbf {A} _{1,1},\ldots ,\mathbf {A} _{m,1},\mathbf {A} _{1,2},\ldots ,\mathbf {A} _{m,2},\ldots ,\mathbf {A} _{1,n},\ldots ,\mathbf {A} _{m,n}]^{\mathrm {T} }$

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,

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

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

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

$\mathbf {K} ^{(r,m)}=\sum _{i=1}^{r}\sum _{j=1}^{m}\left(\mathbf {e} _{r,i}{\mathbf {e} _{m,j}}^{\mathrm {T} }\right)\otimes \left(\mathbf {e} _{m,j}{\mathbf {e} _{r,i}}^{\mathrm {T} }\right).$

Example

Let M be a 2×2 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 4×4 square matrix that will transform vec(M) into vec(M^T)

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

For both square and rectangular matrices of M rows and N columns, the commutation matrix can be generated by this pseudocode, which is similar to an article at Stack Exchange 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 $3\times 2$ matrix $\mathbf {A}$ has two possible vectorizations as follows:

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

and the code above yields

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

giving the expected results

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

$K^{T}\times V_{1}=V_{2}$

$K\times V_{2}=V_{1}$

References

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