Elementary Linear Algebra, 11th Edition

CHAPTER 5Eigenvalues and Eigenvectors

CHAPTER CONTENTS

1. 5.1 Eigenvalues and Eigenvectors
2. 5.2 Diagonalization
3. 5.3 Complex Vector Spaces
4. 5.4 Differential Equations
5. 5.5 Dynamical Systems and Markov Chains

INTRODUCTION

In this chapter we will focus on classes of scalars and vectors known as “eigenvalues” and “eigenvectors,” terms derived from the German word eigen, meaning “own,” “peculiar to,” “characteristic,” or “individual.” The underlying idea first appeared in the study of rotational motion but was later used to classify various kinds of surfaces and to describe solutions of certain differential equations. In the early 1900s it was applied to matrices and matrix transformations, and today it has applications in such diverse fields as computer graphics, mechanical vibrations, heat flow, population dynamics, quantum mechanics, and economics, to name just a few.

5.1 Eigenvalues and Eigenvectors

In this section we will define the notions of “eigenvalue” and “eigenvector” and discuss some of their basic properties.

Definition of Eigenvalue and Eigenvector

We begin with the main definition in this section.

DEFINITION 1

If $A$ is an $n\times n$ matrix, then a nonzero vector $\mathbf{x}$ in $R^n$ is called an eigenvector of $A$ (or of the matrix operator $T_A$) if $A\mathbf{x}$ is a scalar multiple of $\mathbf{x}$; that is,

$$A\mathbf{x}=\mathrm{\lambda }\mathbf{x}$$

for some scalar $\mathrm{\lambda }$. The scalar $\mathrm{\lambda }$ is called an eigenvalue of $A$ (or of $T_A$), and $\mathbf{x}$ is said to be an ...