Kronecker Products and the Vec and Vech Operators

Abstract

Some (though by no means all) of the partitioned matrices encountered in statistics are expressible in terms of two (or more) matrices (of relatively small dimensions) in the form of something called a Kronecker product. In Section 16.1, the definition of a Kronecker product of matrices is given, and a number of results on Kronecker products are presented. These results can (when applicable) be exploited for computational (and other) purposes.

In subsequent sections of this chapter, the notion introduced in Chapter 15 (in connection with the differentiation of a function of a matrix) of rearranging the (nonredundant) elements of a matrix in the form of a column vector is formalized by introducing something called the vec or vech operator. A number of results on the vec or vech operator are presented. Kronecker products appear in many of these results.