Chapter 5 - Eigenvalues and Eigenvectors

Abstract

This chapter defines eigenvalues and their corresponding eigenvectors. It computes the eigenvalues and eigenvectors for a number of examples using polynomial root finding and Gaussian elimination with a homogeneous system. Some properties of eigenvalues are developed, including the fact that an n × n matrix is singular if and only if it has a zero eigenvalue, as well as the fact that the determinant of a matrix is the product of its eigenvalues. The concept of similar matrices is presented, and that similar matrices have the same eigenvalues. This discussion leads to the concept of matrix diagonalization. It is made clear that a matrix can be diagonalized only if it has n linearly independent eigenvectors. If a matrix has distinct eigenvalues, it has n linearly independent eigenvectors. A symmetric matrix can always be diagonalized, even if it has repeated eigenvalues. It is simple to compute powers of a diagonalizable matrix. An application involving an electrical circuit containing batteries, resistors, and inductors is presented, and the solution for the currents involves a first-order differential equation whose solution requires solving an eigenproblem. The chapter discusses irreducible matrices and the Perron-Frobenius theorem. This result is used to develop a scheme for ranking teams. The chapter ends by presenting the MATLAB function eig that computes eigenvalues or eigenvalues and their associated eigenvectors.