### Introduction

In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one or more of the eigenvalues is repeated . We shall see that this sometimes (but not always) causes problems in the diagonalization process that was discussed in the previous Section. We shall then consider the special properties possessed by symmetric matrices which make them particularly easy to work with.

#### Prerequisites

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

#### Learning Outcomes

• state the conditions under which a matrix with repeated eigenvalues may be diagonalized
• state the main properties of real symmetric matrices