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(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:

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

where A = [Ai,j]. In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(AT) is the vector obtaining by vectorizing A in row-major order.

In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator [1]

## Properties

• The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. In particular, K(m,n) is equal to ${\displaystyle \mathbf {P} _{\pi ))$, where ${\displaystyle \pi }$ is the permutation over ${\displaystyle \{1,\dots ,mn\))$ for which
${\displaystyle \pi (i+m(j-1))=j+n(i-1),\quad i=1,\dots ,m,\quad j=1,\dots ,n.}$
• 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,
${\displaystyle \mathbf {K} ^{(r,m)}(\mathbf {A} \otimes \mathbf {B} )\mathbf {K} ^{(n,q)}=\mathbf {B} \otimes \mathbf {A} .}$
This property is often used in developing the higher order statistics of Wishart covariance matrices.[2]
• The case of n=q=1 for the above equation states that for any column vectors v,w of sizes m,r respectively,
${\displaystyle \mathbf {K} ^{(r,m)}(\mathbf {v} \otimes \mathbf {w} )=\mathbf {w} \otimes \mathbf {v} .}$
This property is the reason that this matrix is referred to as the "swap operator" in the context of quantum information theory.
• Two explicit forms for the commutation matrix are 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
${\displaystyle \mathbf {K} ^{(r,m)}=\sum _{i=1}^{r}\sum _{j=1}^{m}\left(\mathbf {e} _{r,i}{\mathbf {e} _{m,j))^{\mathrm {T} }\right)\otimes \left(\mathbf {e} _{m,j}{\mathbf {e} _{r,i))^{\mathrm {T} }\right)=\sum _{i=1}^{r}\sum _{j=1}^{m}\left(\mathbf {e} _{r,i}\otimes \mathbf {e} _{m,j}\right)\left(\mathbf {e} _{m,j}\otimes \mathbf {e} _{r,i}\right)^{\mathrm {T} }.}$
• The commutation matrix may be expressed as the following block matrix:
${\displaystyle \mathbf {K} ^{(m,n)}={\begin{bmatrix}\mathbf {K} _{1,1}&\cdots &\mathbf {K} _{1,n}\\\vdots &\ddots &\vdots \\\mathbf {K} _{m,1}&\cdots &\mathbf {K} _{m,n},\end{bmatrix)),}$
Where the p,q entry of n x m block-matrix Ki,j is given by
${\displaystyle \mathbf {K} _{ij}(p,q)={\begin{cases}1&i=q{\text{ and ))j=p,\\0&{\text{otherwise)).\end{cases))}$
For example,
${\displaystyle \mathbf {K} ^{(3,4)}=\left[{\begin{array}{ccc|ccc|ccc|ccc}1&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&1&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&1&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&1&0&0\\\hline 0&1&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&1&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&1&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&1&0\\\hline 0&0&1&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&1&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&1\end{array))\right].}$

## 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
w = np.arange(m * n).reshape((m, n), order="F").T.ravel(order="F")

# apply this permutation to the rows (i.e. to each column) of identity matrix and return result
return np.eye(m * n)[w, :]


Alternatively, a version without imports:

# Kronecker delta
def delta(i, j):
return int(i == j)

def comm_mat(m, n):
# determine permutation applied by K
v = [m * j + i for i in range(m) for j in range(n)]

# apply this permutation to the rows (i.e. to each column) of identity matrix
I = [[delta(i, j) for j in range(m * n)] for i in range(m * n)]
return [I[i] for i in v]


### MATLAB

function P = com_mat(m, n)

% determine permutation applied by K
A = reshape(1:m*n, m, n);
v = reshape(A', 1, []);

% apply this permutation to the rows (i.e. to each column) of identity matrix
P = eye(m*n);
P = P(v,:);


### R

# Sparse matrix version
comm_mat = function(m, n){
i = 1:(m * n)
j = NULL
for (k in 1:m) {
j = c(j, m * 0:(n-1) + k)
}
Matrix::sparseMatrix(
i = i, j = j, x = 1
)
}


## Example

Let ${\displaystyle A}$ denote the following ${\displaystyle 3\times 2}$ matrix:

${\displaystyle A={\begin{bmatrix}1&4\\2&5\\3&6\\\end{bmatrix)).}$

${\displaystyle A}$ has the following column-major and row-major vectorizations (respectively):

${\displaystyle \mathbf {v} _{\text{col))=\operatorname {vec} (A)={\begin{bmatrix}1\\2\\3\\4\\5\\6\\\end{bmatrix)),\quad \mathbf {v} _{\text{row))=\operatorname {vec} (A^{\mathrm {T} })={\begin{bmatrix}1\\4\\2\\5\\3\\6\\\end{bmatrix)).}$

The associated commutation matrix is

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

(where each ${\displaystyle \cdot }$ denotes a zero). As expected, the following holds:

${\displaystyle K^{\mathrm {T} }K=KK^{\mathrm {T} }=\mathbf {I} _{6))$
${\displaystyle K\mathbf {v} _{\text{col))=\mathbf {v} _{\text{row))}$

## References

1. ^ Watrous, John (2018). The Theory of Quantum Information. Cambridge University Press. p. 94.
2. ^ von Rosen, Dietrich (1988). "Moments for the Inverted Wishart Distribution". Scand. J. Stat. 15: 97–109.
• Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley.