The VEC-Permutation Matrix, the VEC Operator and Kronecker Products: A Review

Abstract

The vec-permutation matrix Im, n is defined by the equation vecAm×n = Im, n vecA', where vec is the vec operator such that vecA is the vector of columns of A stacked one under the other. The variety of definitions, names and notations for Im, n are discussed, and its properties are developed by simple proofs in contrast to certain lengthy proofs in the literature that are based on descriptive definitions. For example, the role of Im, n in reversing the order of Kronecker products is succinctly derived using the vec operator. The matrix Mm, n is introduced as Mm, n= Im, nM; it is the matrix having as rows, every nth row of M, of order mn × c, starting with the first, then every nth row starting with the second, and so on. Special cases of Mm, n are discussed. © 1981, Taylor & Francis Group, LLC. All rights reserved.