### Introduction

Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration analysis, electric circuits, advanced dynamics and quantum mechanics are just a few of the application areas.

Many of the applications involve the use of eigenvalues and eigenvectors in the process of
**
transforming
**
a given matrix into a
**
diagonal
**
matrix and we discuss this process in this Section. We then go on to show how this process is invaluable in solving coupled differential equations of both first order and second order.

#### Prerequisites

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

#### Learning Outcomes

- diagonalize a matrix with distinct eigenvalues using the modal matrix
- solve systems of linear differential equations by the ‘decoupling’ method

#### Contents

1 Diagonalization of a matrix with distinct eigenvalues1.1 Matrix powers

2 Systems of first order differential equations

3 Systems of second order differential equations