Page 194 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 16: Direct Sum Decompositions of
Elementary Canonical Forms

CONTENTS
Objectives
Introduction

16.1 Overview
16.2 Direct-sum Decompositions
16.3 Summary
16.4 Keywords

16.5 Review Questions
16.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the meanings of invariant subspaces as well as decomposition of a vector
space into the invariant direct sums of the independent subspaces.
 Know the projection operators and their properties

 See that there is less emphasis is on matrices and more attention is given to subspaces.
Introduction

This unit and the next units are slightly more complicated than the other previous units.
The ideas of invariant subspaces and their relations with the vector space V is obtained.

The ideas of projection operators and their properties are introduced. These ideas will help in
expressing the given linear operator T in terms of the direct sums of the operators T T as seen
1j K
in the next unit.
16.1 Overview

We are again going to analyse a single linear operator on a finite dimensional space V over the
field F. In the next three units we shall stress less in terms of matrices and stress more on the
subspaces, in order to find an ordered basis in which the matrix of T assumes an especially a
simple form. Our aim in three units will be as follows: To decompose the underlying space V
into a sum of invariant subspaces for T such that the restriction operators on these subspaces are
simple. These subspaces will be taken as independent subspaces of the vector space V and after
finding the independent basis of each independent subspace the ordered basis of the whole
space will be constructed. Given such a decomposition of the vector space we then see that T
induces a linear operator T on each subspace W , by restriction. We shall describe this situation
i i
by saying that the linear operator is the invariant direct sum of the operators T , T ,..., T . Once
1 2 k
the space is decomposed in terms of invariant subspaces, we shall introduce the concepts of
projection operators on V.

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