## 2

## Eigenvalues and Eigenvectors

#### 2.1 Introduction

In this chapter we will consider polynomials whose coefficients are not real numbers but *n*-square matrices, ﬁnd eigenvalues and eigenvectors of a matrix, state and prove an important theorem known as the Cayley–Hamilton Theorem and see how it can be used to ﬁnd powers of square matrices and inverses of nonsingular matrices.

#### 2.1.1 Matrix Polynomial

If *P*_{0}, *P*_{1}, …, *P*_{m} (≠ 0) are *n*-square matrices, then an expression of the form

is called a matrix polynomial of degree ‘*m*’.

*E.g.*

Then

is a matrix ...