Page 255 - Advanced Linear Algebra

Structure Theory for Normal Operators 239

are real. This would suggest that the analog of the set of real numbers is the set
of self-adjoint matrices. Also, we will see that a complex matrix is unitary if and
only if its eigenvalues have norm , so numbers on the unit circle seem to

correspond to the set of unitary matrices. This leaves open the question of which
normal matrices correspond to the positive real numbers. These are the positive
definite matrices, which we will discuss later in the chapter.
Self-Adjoint Operators

Let us consider the basic properties of self-adjoint operators. The quadratic

form associated with the linear operator is the function 8¢ = ¦ - defined

by

8 ²#³ ~ º #Á #»

We have seen (Theorem 9.2) that in a complex inner product space, ~ if and
only if 8~ but this does not hold, in general, for real inner product spaces.

However, it does hold for symmetric operators on a real inner product space.
Theorem 10.11 Let = be a finite-dimensional inner product space and let
Á B²= ³.
)
1 If and are self-adjoint, then so are the following:

a ) b
b ) c , if is invertible

)
c ² ³ , for any real polynomial ²%³  ´%µ
s

)

2 A complex operator is Hermitian if and only if 8²#³ is real for all

#= .
3 If is a complex operator or a real symmetric operator, then
)

~ ¯ 8 ~

)

4 The characteristic polynomial ²%³ of a self-adjoint operator splits over

s, that is, all complex roots of ²%³ are real. Hence, the minimal

polynomial ²%³ of is the product of distinct monic linear factors over

s.
Proof. For part 2), if is Hermitian, then

º #Á #» ~ º#Á #» ~ º #Á #»

and so 8 ²#³ ~ º #Á #» is real. Conversely, if º #Á #»  s , then

º#Á #» ~ º #Á #» ~ º#Á i #»

and so ~ i .

For part 3), we need only prove that 8~ implies ~ when - ~ s . But if

8~ , then