Commutation matrix

Not to be confused with the commutator A B − B A {\displaystyle AB-BA} of two square matrices A {\displaystyle A} and B {\displaystyle B} in ring theory.

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 permutation 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:

vec ⁡ ( A ) = [ A 1 , 1 , … , A m , 1 , A 1 , 2 , … , A m , 2 , … , A 1 , n , … , A m , n ] T {\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. The cycles and other properties of this permutation have been heavily studied for in-place matrix transposition algorithms.

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

• The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. In particular, K^(m,n) is equal to P π {\displaystyle \mathbf {P} _{\pi }} , where π {\displaystyle \pi } is the permutation over { 1 , … , m n } {\displaystyle \{1,\dots ,mn\}} for which

π ( i + m ( j − 1 ) ) = j + n ( i − 1 ) , i = 1 , … , m , j = 1 , … , n . {\displaystyle \pi (i+m(j-1))=j+n(i-1),\quad i=1,\dots ,m,\quad j=1,\dots ,n.}

• The determinant of K^(m,n) is ( − 1 ) 1 4 n ( n − 1 ) m ( m − 1 ) {\displaystyle (-1)^{{\frac {1}{4}}n(n-1)m(m-1)}} .
• 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,

K ( r , m ) ( A ⊗ B ) K ( n , q ) = B ⊗ A . {\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,

K ( r , m ) ( v ⊗ w ) = w ⊗ v . {\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 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

K ( r , m ) = ∑ i = 1 r ∑ j = 1 m ( e r , i e m , j T ) ⊗ ( e m , j e r , i T ) = ∑ i = 1 r ∑ j = 1 m ( e r , i ⊗ e m , j ) ( e m , j ⊗ e r , i ) T . {\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:

K ( m , n ) = [ K 1 , 1 ⋯ K 1 , n ⋮ ⋱ ⋮ K m , 1 ⋯ K m , n , ] , {\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

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

For example,

K ( 3 , 4 ) = [ 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 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 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 ] . {\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].}

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

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]

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,:);

# 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
  )
}

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

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

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

v col = vec ⁡ ( A ) = [ 1 2 3 4 5 6 ] , v row = vec ⁡ ( A T ) = [ 1 4 2 5 3 6 ] . {\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

K = K ( 3 , 2 ) = [ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ] , {\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:

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

K v col = v row {\displaystyle K\mathbf {v} _{\text{col}}=\mathbf {v} _{\text{row}}}

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