Some results on commutation matrices, with statistical applications

The commutation matrix Pmn changes the order of multiplication of a Kronecker matrix product. The vec operator stacks columns of a matrix one under another in a single column. It is possible to express the vec of a Kronecker matrix product in terms of a Kronecker product of vecs of matrices. The commutation matrix plays an important role here. “Super‐vec‐operators” like vec A ⊗ vec A vec (A ⊗ A), and vec{(A ⊗ A)Pnn} are very convenient. Several of their properties are being studied. Both the traditional commutation matrix and vec operator and the newer concepts developed from these are applied to multivariate statistical and related problems.