Eigenvalues and Eigenvectors

Introduction

From an applications viewpoint, eigenvalue problems are probably the most important problems that arise in connection with matrix analysis. In this Section we discuss the basic concepts. We shall see that eigenvalues and eigenvectors are associated with square matrices of order $n \times n$ . If $n$ is small (2 or 3), determining eigenvalues is a fairly straightforward process (requiring the solutiuon of a low order polynomial equation). Obtaining eigenvectors is a little strange initially and it will help if you read this preliminary Section first.

Prerequisites

have a knowledge of determinants and matrices
have a knowledge of linear first order differential equations

Learning Outcomes

obtain eigenvalues and eigenvectors of $2 \times 2$ and $3 \times 3$ matrices
state basic properties of eigenvalues and eigenvectors

1 Basic concepts

1.1 Determinants

1.2 Systems of linear equations

1.3 Basic results in linear algebra

1.4 A simple eigenvalue problem

2 General eigenvalue problems

2.1 Case 1

3 Properties of eigenvalues and eigenvectors