ABSTRACT
In this chapter we introduce the fundamental concepts of eigenvalues and eigenvectors of a square matrix in the classical way. The eigenvalues of a matrix A of order n are roots of a polynomial, called the characteristic polynomial of A. The coefficients of this polynomial are sums of certain determinants of submatrices of A and thus can be described using digraphs as shown in Section 7.1. In Section 7.2, we give a combinatorial argument for the Cayley-Hamilton theorem, which asserts that a matrix satisfies its characteristic polynomial. The study of eigenvalues leads to the notion of similarity of matrices and this, in turn, leads to the Jordan Canonical Form of a matrix in Section 7.3. We give a highly combinatorial argument for the existence of the Jordan Canonical Form of a matrix. The chapter is concluded with Section 7.4 which describes how eigenvalues of circulants, introduced in Chapter 3, can be calculated using associated digraphs.
