The vec-permutation matrix, the vec operator and Kronecker products: a review

Abstract The vec-permutation matrix I m,n is defined by the equation vec A m × n = I m,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 I m,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 I m,n in reversing the order of Kronecker products is succinctly derived using the vec operator. The matrix M m,n is introduced as M m,n = I m,n M; it is the matrix having as rows,every nth row starting with the first, then every nth row starting with the second, and so on. Special cases of M m,n are discussed. † Supported by a New Zealand National Research Advisory Council Research Fellowship during his sojourn at Cornell. † Supported by a New Zealand National Research Advisory Council Research Fellowship during his sojourn at Cornell. Notes † Supported by a New Zealand National Research Advisory Council Research Fellowship during his sojourn at Cornell.