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. 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 a triangular matrix. But there are other matrix representations that reflect intrinsic properties of linear operators. This chapter deals with some such representations usually known as the canonical forms.
To motivate our approach, recall that (see discussion after Proposition 5.5.5) a linear operator T on a finite-dimensional ...
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