Page 183 - DMTH502_LINEAR_ALGEBRA

Richa Nandra, Lovely Professional University Unit 14: Invariant Subspaces

Unit 14: Invariant Subspaces Notes

CONTENTS
Objectives
Introduction

14.1 Invariant Subspaces: Definitions
14.2 Theorems and Examples
14.3 Summary

14.4 Keywords
14.5 Review Questions
14.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Know about few concepts which are useful in analysing further properties of the linear
operator T.
 Understand concepts like invariant subspace, the restriction operator Tw, the T-conductor
of a vector into subspace W.
 See that all these concepts help us in understanding the structure of minimal polynomial
of linear operator.
 Understand the restriction operator Tw helps in writing the matrix A of the linear operator
in a block form and so the characteristic polynomial for Tw divides the characteristic
polynomial for T.

Introduction

In this unit we are still studying the properties of a linear operator on the vector space V. The
concept of invariant subspace, the restriction operator Tw help us in finding the characteristic
polynomial of T as well as its annihilator and so it helps in diagonalization of the matrix A of the
linear operator T.
14.1 Invariant Subspaces: Definitions

In this unit, we shall introduce a few concepts which are useful in analysing further the properties
of a linear operator. We shall use these concepts to obtain characterizations of diagonalizable
(and triangulable) operators in terms of their minimal polynomials.

Invariant Subspace

A subspace W of the vector space V is invariant of more precisely T-invariant if for each vector
 in W the vector T is in W, i.e., T(w) is contained in W. When this is so T induces a linear
operator on W, called restriction to W. We often denote the restriction by Tw. The linear operator
Tw is defined by Tw() = T(), for  in W, but Tw is quite a different object from T since its
domain is W and not V.

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