INTRODUCTION

In this chapter, we consider matrix operators that are used throughout the book and special square matrices, namely triangular matrices and band matrices, that will crop up continually in our future work. From the elements of an m x n matrix, A = (aij) and a p x q matrix, B = (bij), the Kronecker product forms an mp x nq matrix. The vec operator forms a column vector out of a given matrix by stacking its columns one underneath the other. The devec operator forms a row vector out of a given matrix by stacking its rows one alongside the other. In like manner, a generalized vec operator forms a new matrix from a given matrix by stacking a certain number of its columns under each other and a generalized devec operator forms a new matrix by stacking a certain number of rows alongside each other. It is well known that the Kronecker product is intimately connected with the vec operator, but we shall see that this connection also holds for the devec and generalized operators as well. Finally we look at special square matrices with zeros above or below the main diagonal or whose nonzero elements form a band surrounded by zeros.