Linear Algebra

6

Canonical Forms

6.1 INTRODUCTION

In Chapter 5, we saw that a linear operator on a finite-dimensional vector space can be diagonalized only under some strict conditions on its minimal or characteristic polynomial. As we saw in some examples, even if the operator has eigenvalues, it may fail to be diagonalizable. Thus, the class of diagonalizable operators is rather limited. So, we seek other simple forms of matrix representations of linear operators. An upper or a lower triangular matrix is an example of such simple forms, and we have already seen that over ℂ, any operator can be represented as an upper triangular matrix. But there are other matrix representations that reflect intrinsic properties of linear operators. This chapter deals with ...