Page 311 - DMTH502_LINEAR_ALGEBRA

Sachin Kaushal, Lovely Professional University Unit 29: Spectral Theory and Properties of Normal Operators

Unit 29: Spectral Theory and Properties of Notes
Normal Operators

CONTENTS
Objectives
Introduction

29.1 Spectral Theory
29.2 Properties of Normal Operators
29.3 Summary
29.4 Keywords

29.5 Review Questions
29.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand that Theorems 9 and 13 of unit 26 are pursued further concerning the
diagonalization of self-adjoint and normal operators.
 See that if T is a normal operator or a self-adjoint operator on a finite dimensional inner
product space V. Let C , C be the distinct characteristic values of T and W be the
1 k i
characteristic space associated with C and E be the orthogonal projection of V on W , then
i i i
V is the direct sum of W , W , ... W and T = C E + C E + ... + C E which is called spectral
1 2 k 1 1 2 2 k k
resolution of T.
 See that if A is a normal matrix with real (complex) entries, then there is a real orthogonal
–1
(unitary) matrix P such that P AP is in rational canonical form.
Introduction

In this unit the properties of the normal operators or the self-adjoint operator are studied
further.
The spectral resolution of the linear operator T is given by the decomposition T = C E + C E +
1 1 2 2
E C , where C , C ... C are the distinct characteristic values of T and E , E ... E are the orthogonal
k k 1 2 k 1 2 k
projections of V on W , W ... W .
1 2 k
If T is a diagonalizable normal operator on a finite dimensional inner product space V, then T is
self-adjoint, non-negative or unitary according as each characteristic value of T is real,
non-negative or of absolute value 1.

The family of orthogonal projections (P , P , ... P ) is called the resolution of the identity determined
1 2 k
( )P
bF, and T = r T j is the spectral resolution of T in terms of this family.
j
j

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