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].

In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator

Properties

• 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,

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

This property is often used in developing the higher order statistics of Wishart covariance matrices.

• The case of n=q=1 for the above equation states that for any column vectors v,w of sizes m,r respectively,

$\mathbf {K} ^{(r,m)}(\mathbf {v} \otimes \mathbf {w} )=\mathbf {w} \otimes \mathbf {v} .$

This property is the reason that this matrix is referred to as the "swap operator" in the context of quantum information theory.

• 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

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

• The commutation matrix may be expressed as a the following block matrix:

$\mathbf {K} ^{(m,n)}={\begin{bmatrix}\mathbf {K} _{1,1}&\cdots &\mathbf {K} _{1,n}\\\vdots &\ddots &\vdots \\\mathbf {K} _{m,1}&\cdots &\mathbf {K} _{m,n},\end{bmatrix}},$

Where the p,q entry of n x m block-matrix K_i,j is given by

$\mathbf {K} _{ij}(p,q)={\begin{cases}1&i=q{\text{ and }}j=p\\0&{\text{otherwise}}.\end{cases}}$

For example,

$\mathbf {K} ^{(3,4)}=\left[{\begin{array}{ccc|ccc|ccc|ccc}1&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&1&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&1&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&1&0&0\\\hline 0&1&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&1&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&1&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&1&0\\\hline 0&0&1&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&1&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&1\end{array}}\right].$

Code

For both square and rectangular matrices of m rows and n columns, the commutation matrix can be generated by the code below.

Python

import numpy as np

def comm_mat(m,n):
    # determine permutation applied by K
    v = np.arange(m*n)
    A = np.reshape(v,(m,n),order = 'F')
    w = np.reshape(A.T,m*n, order = 'F')

    # apply this permutation to the rows (i.e. to each column) of K
    M = np.eye(m*n)
    M = M[:,w]
    return M

Example

Let M be a 2×2 square matrix.

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

Then we have

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

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

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

The $3\times 2$ matrix $A$ has two possible vectorizations as follows:

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

and the code above yields

$K=\mathbf {K} ^{(3,2)}={\begin{bmatrix}1&\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &1&\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &1&\cdot \\\cdot &1&\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &1&\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &\cdot &1\\\end{bmatrix}}$

giving the expected results

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

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

$KV_{2}=V_{1}$

References

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