Английская Википедия: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(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:
- <math>\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}}</math>
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 <math>\mathbf P_\pi</math>, where <math>\pi</math> is the permutation over <math>\{1,\dots,mn\}</math> for which
- <math>
\pi(i + m(j-1)) = j + n(i-1), \quad i = 1,\dots,m, \quad j = 1,\dots,n. </math>
- Replacing A with AT in the definition of the commutation matrix shows that Шаблон:Nowrap 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,
- <math>\mathbf{K}^{(r, m)} (\mathbf{A} \otimes \mathbf{B}) \mathbf{K}^{(n, q)} = \mathbf{B} \otimes \mathbf{A}.</math>
- 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,
- <math> \mathbf{K}^{(r,m)}(\mathbf v \otimes \mathbf w) = \mathbf w \otimes \mathbf v.</math>
- 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
- <math>\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}} .</math>
- The commutation matrix may be expressed as the following block matrix:
- <math>
\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}, </math>
- Where the p,q entry of n x m block-matrix Ki,j is given by
- <math>
\mathbf K_{ij}(p,q) = \begin{cases} 1 & i=q \text{ and } j = p,\\ 0 & \text{otherwise}. \end{cases} </math>
- For example,
- <math>
\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]. </math>
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 <math>A</math> denote the following <math>3 \times 2</math> matrix:
- <math>
A = \begin{bmatrix}
1 & 4 \\ 2 & 5 \\ 3 & 6 \\
\end{bmatrix}. </math> <math>A</math> has the following column-major and row-major vectorizations (respectively):
- <math>
\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}. </math>
The associated commutation matrix is
- <math> 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},
</math> (where each <math>\cdot</math> denotes a zero). As expected, the following holds:
- <math> K^\mathrm{T} K = KK^\mathrm{T} =\mathbf I_6 </math>
- <math> K \mathbf v_{\text{col}} = \mathbf v_{\text{row}} </math>
References
- Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley.
