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

The vee-permutation matrix I is defined by the equation _m,n vecAmX =I vecA', where vee is the vee operator such that vecA,.. n _m, n,.. is the vector of columns of A stacked one under the other. The variety of definitions, names and notations for I are discussed,,..m, n and its properties are developed by simple proofs in contrast to certain lengthy proofs in the literature that are based on descrip-tive definitions. For example, the role of I in reversing the,..m,n order of Kronecker products is succinctly derived using the vee operator. The matrix M is introduced as M = I M; it is the,..m, n,..m, n,..m, n,.. matrix having for rows, every n'th row of M, of order mn X c, starting with the first, then every n'th row starting with the second, and so on. Special cases of M are discussed