Page 282 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 26: Unitary Operators and Normal Operators

CONTENTS
Objectives
Introduction

26.1 Unitary Operators
26.2 Normal Operators
26.3 Summary

26.4 Keywords
26.5 Review Questions
26.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the meaning of unitary operators, i.e. a unitary operator on an inner product
space is an isomorphism of the space onto itself.
 See that unitary and orthogonal matrices are explained with the help of some examples.

 Understand that for each invertible n × n matrix B in the general linear group GL (n) there
exist unique unitary matrix U and lower triangular matrix M such that U = MB.
 Know that the linear operator T is normal if it commutes with its adjoint TT* = T*T.
–1
 Understand that for every normal matrix A there is a unitary matrix P such that P AP is a
diagonal matrix.

Introduction

In this unit there are two sections – one dealing with unitary operators on finite dimensional
inner product spaces and other dealing with the normal operators.
It is shown that if an n × n matrix B belongs to GL (n) then there exist unique matrices N and U
such that N is in T (n), U is in U (n), and B = N.U.
+
In the second section properties of normal operators are studied. It is seen that a complex n × n
matrix A is said to be normal if A*A = AA*.

With the help of some theorems it is shown that for a normal operator T on V, a finite dimensional
complex inner product space, V has an orthonormal basis consisting of characteristic vectors
for T.

26.1 Unitary Operators

In this unit we first of all consider the concept of an isomorphism between two inner product
spaces. An isomorphism of two vector spaces V onto W is a one-one linear transformation from
V onto W. Now an inner product space consists of a vector space and a specified inner product on
that space. Thus, when V and W are inner product spaces, we shall require an isomorphism from

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