### Introduction

In Section 22.1 it was shown how to obtain eigenvalues and eigenvectors for low order matrices, $2\times 2$ and $3\times 3$ . This involved firstly solving the characteristic equation det $\left(A-\lambda I\right)=0$ for a given $n\times n$ matrix $A$ . This is an ${n}^{\text{th}}$ order polynomial equation and, even for $n$ as low as 3, solving it is not always straightforward. For large $n$ even obtaining the characteristic equation may be difficult, let alone solving it.

Consequently in this Section we give a brief introduction to alternative methods, essentially
**
numerical
**
in nature, of obtaining eigenvalues and perhaps eigenvectors.

We would emphasize that in some applications such as Control Theory we might only require one eigenvalue of a matrix
$A$
, usually the one largest in magnitude which is called the
**
dominant
**
eigenvalue. It is this eigenvalue which sometimes tells us how a control system will behave.

#### Prerequisites

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

#### Learning Outcomes

- use the power method to obtain the dominant eigenvalue (and associated eigenvector) of a matrix
- state the main advantages and disadvantages of the power method

#### Contents

1 Numerical determination of eigenvalues and eigenvectors1.1 Preliminaries

1.2 The power method

1.3 Problems with the power method

1.4 Advantages of the power method

1.5 Finding eigenvalues other than the dominant