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:

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

where A = [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 ^[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

${\displaystyle \pi (i+mj)=j+ni,\quad i=1,\dots ,m,\quad j=1,\dots ,n.}$

• 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,

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

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.

• 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

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

• 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 K_i,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
    A = np.reshape(np.arange(m*n),(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 identity matrix
    M = np.eye(m*n)
    M = M[w,:]
    return M

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
    M = [[delta(i,j) for j in range(m*n)] for i in range(m*n)]
    M = [M[i] for i in v]
    return M

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 ${\displaystyle A}$ denote the following ${\displaystyle 2\times 3}$ 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[]

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