Page 243 - Advanced Linear Algebra

Chapter 10

Structure Theory for Normal Operators

Throughout this chapter, all vector spaces are assumed to be finite-dimensional
unless otherwise noted. Also, the field is either or .
s
-
d
The Adjoint of a Linear Operator
The purpose of this chapter is to study the structure of certain special types of
linear operators on finite-dimensional real and complex inner product spaces. In
order to define these operators, we introduce another type of adjoint (different
from the operator adjoint of Chapter 3).

Theorem 10.1 Let and > be finite-dimensional inner product spaces over -
=

B
and let ²= Á > ³ . Then there is a unique function i ¢ > ¦ = , defined by
the condition
º#Á $» ~ º#Á i $»

for all #= and $> . This function is in ²> Á = ³ and is called the adjoint
B
of .

Proof. If exists, then it is unique, for if
i

º#Á $» ~ º#Á $»
#
then º#Á $» ~ º#Á i $» for all and and so ~ i .
$

i
We seek a linear map ¢> ¦ = for which

º#Á i $» ~ º #Á $»
i
By way of motivation, the vector $ , if it exists, looks very much like a linear
#
map sending to º # Á $ » . The only problem is that i # is supposed to be a
vector, not a linear map. But the Riesz representation theorem tells us that linear
maps can be represented by vectors.