Eigenvalues and Eigenvectors

This chapter discusses the concepts eigenvalues and eigenvectors. For any square matrix A over F , the polynomial det( A — xI ) is the characteristic polynomial of A . The equation det( A — xI ) = 0 is the characteristic equation of A , and the solutions of det( A — xI ) = 0 are called the eigenvalues of A . The set of all eigenvalues of A is called the spectrum of A . In the chapter, T denotes a field, V denotes a finite-dimensional vector space over T , and T denotes a linear operator on V . The chapter discusses the important connection between the eigenvalues of linear operators and those of matrices. If the n × n matrix A represents T relative to the basis A of V , then X is an eigenvector of A corresponding to λ only if X is the coordinate matrix relative to A of an eigenvector of T corresponding to the same eigenvalue.