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

Inmathematics, especially inlinear algebra andmatrix theory, thecommutation matrix is used for transforming thevectorized form of amatrix into the vectorized form of itstranspose. Specifically, the commutation matrixK^(m,n) is thenm ×mnpermutation matrix which, for anym ×n matrixA, transforms vec(A) into vec(A^T):

K^(m,n) vec(A) = vec(A^T) .

Here vec(A) is themn × 1column vector obtain by stacking the columns ofA on top of one another:

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

whereA = [A_i,j]. In other words, vec(A) is the vector obtained by vectorizingA incolumn-major order. Similarly, vec(A^T) is the vector obtaining by vectorizingA in row-major order. Thecycles and other properties of this permutation have been heavily studied forin-place matrix transposition algorithms.

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

• The commutation matrix is a special type ofpermutation matrix, and is thereforeorthogonal. 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

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

• The determinant ofK^(m,n) is${\displaystyle (-1{)}^{\frac{1}{4}n(n-1)m(m-1)}}$.
• ReplacingA withA^T in the definition of the commutation matrix shows thatK^(m,n) = (K^(n,m))^T. Therefore, in the special case ofm =n the commutation matrix is aninvolution andsymmetric.
• The main use of the commutation matrix, and the source of its name, is to commute theKronecker product: for everym ×n matrixA and everyr ×q matrixB,

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

This property is often used in developing the higher order statistics of Wishart covariance matrices.^[2]

• The case ofn=q=1 for the above equation states that for any column vectorsv,w of sizesm,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.

• Two explicit forms for the commutation matrix are as follows: ife_r,j denotes thej-th canonical vector of dimensionr (i.e. the vector with 1 in thej-th coordinate and 0 elsewhere) then

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

• The commutation matrix may be expressed as the following block matrix:

Where thep,q entry ofn x m block-matrixK_i,j is given by

For example,

For both square and rectangular matrices ofm rows andn columns, the commutation matrix can be generated by the code below.

importnumpyasnpdefcomm_mat(m,n):# determine permutation applied by Kw=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 resultreturnnp.eye(m*n)[w,:]

Alternatively, a version without imports:

# Kronecker deltadefdelta(i,j):returnint(i==j)defcomm_mat(m,n):# determine permutation applied by Kv=[m*j+iforiinrange(m)forjinrange(n)]# apply this permutation to the rows (i.e. to each column) of identity matrixI=[[delta(i,j)forjinrange(m*n)]foriinrange(m*n)]return[I[i]foriinv]

functionP=com_mat(m, n)% determine permutation applied by KA=reshape(1:m*n,m,n);v=reshape(A',1,[]);% apply this permutation to the rows (i.e. to each column) of identity matrixP=eye(m*n);P=P(v,:);

# Sparse matrix versioncomm_mat=function(m,n){i=1:(m*n)j=NULLfor(kin1:m){j=c(j,m*0:(n-1)+k)}Matrix::sparseMatrix(i=i,j=j,x=1)}

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

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

${\displaystyle {\mathbf{v}}_{\text{col}}=\mathrm{vec}(A)=\left[\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}\right],{\mathbf{v}}_{\text{row}}=\mathrm{vec}(A^{\mathrm{T}})=\left[\begin{array}{c}1\\ 4\\ 2\\ 5\\ 3\\ 6\end{array}\right].}$

The associated commutation matrix is

(where each${\displaystyle \cdot }$ denotes a zero). As expected, the following holds:

${\displaystyle K^{\mathrm{T}}K=K{K}^{\mathrm{T}}={\mathbf{I}}_6}$

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

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

• Linear algebra
• Matrices

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