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Sachin Kaushal, Lovely Professional University Unit 17: Invariant Direct Sums

Unit 17: Invariant Direct Sums Notes

CONTENTS
Objectives
Introduction

17.1 Overview
17.2 Some Theorems
17.3 Summary

17.4 Keywords
17.5 Review Questions
17.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that the vector space V is decomposed as a direct sum of the invariant subspaces under
some linear operator T.
 Understand that the linear operator induces a linear operator T on each invariant subspaces
i
W by restriction.
i
 Know that if  is the vector in the invariant subspace W then the vector  in the finite
i i
vector space V is obtained as a linear combinations of its projections  in the subspace W .
i i
Introduction

In this unit we again consider a linear transformation T on the finite vector space. Here the
vector space is decomposed as the direct sum of the invariant subspaces W . The linear operator
i
induces a linear operator T for each invariant subspaces W .
i i
The method of finding the projection operators and their properties is discussed.

17.1 Overview

In this unit we are primarily interested in the direct sum decomposition V = W  W  + ... + W ,
1 2 K
where each of the subspaces W is invariant under some linear operator T. Given such a
i
decomposition of V, T induces a linear operator T on each W by restriction. If  is the vector in
i i i
W then the vector  in V can be given as a linear combinations of its projection  in the invariant
i i
subspace W . Thus the action of T is then understood as follows:
i
If  is a vector in V, we have unique vectors  ,...,  with  in W such that
1 k i i
 =  + ... + 
1 k
and then
T = T  + ... + T 
1 1 k k
We shall describe this situation by saying that T is the direct sum of the operators T ,..., T . It must
1 k
be remembered in using this terminology that the T are not linear operators on the space V but
i
on the various subspaces W . The fact that V = W  ...  W enables us to associate with each  in
i 1 k
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