Page 161 - DMTH502_LINEAR_ALGEBRA

Richa Nandra, Lovely Professional University Unit 12: Introduction and Characteristic Values of Elementary Canonical Forms

Unit 12: Introduction and Characteristic Values of Notes
Elementary Canonical Forms

CONTENTS
Objectives
Introduction

12.1 Overview
12.2 Characteristic Values
12.3 Summary
12.4 Keywords

12.5 Review Questions
12.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Know that when the matrix of the linear transformation is in the diagonal form for some
ordered basis the properties of the transformation can be seen at a glance.
 See that a matrix A of a linear operator T can be cast into a diagonal form under similarity
transformations.
–1
 See that a matrix A and P AP where P is an invertible have the same characteristic values.
Introduction

In this unit it is shown how a matrix has a diagonal form.

For this purpose the characteristic values and characteristic vectors are worked out and an
invertible matrix is worked out of the characteristic vectors that can diagonalize the given
matrix.

12.1 Overview

One of our primary aim in these units is to study linear transformation on finite dimensional
vector spaces. So far we have studied many specific properties of linear transformations. In
terms of ordered basis vectors we have represented such types of matrices by matrices. In terms
of matrices we see lots of insight of the linear transformation. We also explored the linear
algebra L(V, V) consisting of the linear transformations of a space into itself.
In the next few units we shall concentrate ourselves with linear operators on a finite dimensional
vector space. If we consider the ordered basis = ( , ,... ) then the effect of T on is
1 2 n i
n
T A
i ji j i = 1, 2, ... n
j 1
where the new ordered basis is = ( , , ... ) . If we now choose the basis = ( , , ... ) in
1 2 n 1 2 n
such a way that

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