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(\mathbf) = mathbf_, \ldots, \mathbf_, \mathbf_, \ldots, \mathbf_, \ldots, \mathbf_, \ldots, \mathbf_
where A = ''A_''i'',''j'' In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(A^T) 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

Properties

* The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. In particular, K^{(''m'',''n'')} is equal to $\mathbf P_\pi$, where $\pi$ is the permutation over $\$ for which
::$\pi(i + m(j-1)) = j + n(i-1), \quad i = 1,\dots,m, \quad j = 1,\dots,n.$

* Replacing A with A^T in the definition of the commutation matrix shows that 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,
::$\mathbf^ (\mathbf \otimes \mathbf) \mathbf^ = \mathbf \otimes \mathbf.$
: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,
::$\mathbf^(\mathbf v \otimes \mathbf w) = \mathbf w \otimes \mathbf v.$
: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 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
::$\mathbf^ = \sum_^r \sum_^m \left(\mathbf_ ^\right) \otimes \left(\mathbf_ ^\right) = \sum_^r \sum_^m \left(\mathbf_ \otimes \mathbf_\right) \left( \mathbf_ \otimes \mathbf_\right)^ .$

* The commutation matrix may be expressed as the following block matrix:
::$\mathbf^ = \begin \mathbf_ & \cdots & \mathbf_\\ \vdots & \ddots & \vdots\\ \mathbf_ & \cdots & \mathbf_, \end,$
:Where the ''p,q'' entry of ''n x m'' block-matrix K_''i,j'' is given by
::$\mathbf K_(p,q) = \begin 1 & i=q \text j = p,\\ 0 & \text. \end$
:For example,
::$\mathbf K^ = \left[\begin 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end\right].$

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) , :

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 = * 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 [ifor i in v">[i.html" ;"title="[i">[ifor 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,:);

Example

Let $A$ denote the following $3 \times 2$ matrix:
:$A = \begin 1 & 4 \\ 2 & 5 \\ 3 & 6 \\ \end.$
$A$ has the following column-major and row-major vectorizations (respectively):

:$\mathbf v_ = \operatorname(A) = \begin 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ \end , \quad \mathbf v_ = \operatorname(A^) = \begin 1 \\ 4 \\ 2 \\ 5 \\ 3 \\ 6 \\ \end.$

The associated commutation matrix is

:$K = \mathbf K^ = \begin 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 1 & \cdot & \cdot \\ \cdot & 1 & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 1 & \cdot \\ \cdot & \cdot & 1 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & 1 \\ \end,$
(where each $\cdot$ denotes a zero). As expected, the following holds:
:$K^\mathrm K = KK^\mathrm =\mathbf I_6$
:$K \mathbf v_ = \mathbf v_$

References

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

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