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