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):
Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another:
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]
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 denote the following matrix:
has the following column-major and row-major vectorizations (respectively):
The associated commutation matrix is
(where each denotes a zero). As expected, the following holds: