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 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:

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

Properties

• The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. In particular, K^(m,n) is equal to , where is the permutation over for which

:

• The determinant of K^(m,n) is .
• Replacing A with A^T in the definition of the commutation matrix shows that 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,

:

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

• The case of n=q=1 for the above equation states that for any column vectors v,w of sizes m,r respectively,

:

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

:

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

:

Where the p,q entry of n x m block-matrix K_i,j is given by

:

For example,

:

Code

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

Python

Alternatively, a version without imports:

MATLAB

R

Example

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:

References

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