Chapter 6 - MATRIX CALCULUS

Abstract

This chapter discusses matrix calculus. It discusses matrix functions, specifically polynomials and exponentials, developing techniques for calculating these functions, and some of their important properties. As a matrix commutes with itself, many of the properties of polynomials—addition, subtraction, multiplication, and factoring but not division—are still valid for polynomials of a matrix. The chapter presents one of the most powerful theorems of matrix theory, the Cayley–Hamilton theorem. In general, it is very difficult to compute the functions of matrices from their definition as infinite series—one exception is the diagonal matrix. The Cayley–Hamilton theorem, however, provides a starting point for the development of an alternate, straightforward method for calculating these functions. The chapter also presents the method for the polynomials of matrices having distinct eigenvalues. The method is extended to the functions of matrices having arbitrary eigenvalues.