Trace of a (Square) Matrix

Abstract

The primary subject of this relatively short chapter is a basic characteristic of a square matrix called the trace. There are many areas of statistics in which the trace of a matrix is encountered. For example, in the design of experiments, one of the criteria (A-optimality) that is used for comparing designs involves the trace of a matrix (Fedorov 1972).

The trace of a matrix is defined and its basic properties described in Section 5.1. Some very useful results on the trace of a product of matrices are covered in Section 5.2. One of these results gives rise to some very useful mathematical equivalences, which are presented in Section 5.3. That the results on the trace of a matrix and the related equivalences are placed in a separate chapter is indicative not only of their importance but of the fact that they don’t fit particularly well into any of the other chapters.