Page 250 - Advanced Linear Algebra

234 Advanced Linear Algebra

)
3 = has the form
=~ ; p Ä p ;

where ÁÃÁ are the distinct eigenvalues of .
For simplicity in exposition, we will tend to use the term unitarily
diagonalizable for both cases. Since unitarily diagonalizable operators are so
well behaved, it is natural to seek a characterization of such operators.
Remarkably, there is a simple one, as we will see next.
Normal Operators

Operators that commute with their own adjonts are very special.
Definition
)
1 A linear operator on an inner product space is normal if it commutes
=

with its adjoint:
i
i
~
) is normal i
(
2 A matrix ( C ²-³ if commutes with its adjoint ( .
If is normal and is an ordered orthonormal basis of , then

=
E
i
i

´µ ´µ ~ ´µ ´ µ ~ ´ i µ E

E
E
E
E
and
i
i
´µ ´µ ~ ´ µ ´µ ~ ´ i

E E E E µ E
and so is normal if and only if ´µ E is normal for some, and hence all,

orthonormal bases for . Note that this does not hold for bases that are not
=
orthonormal.
Normal operators have some very special properties.
Theorem 10.8 Let ²= ³ be normal.

B
)
1 The following are also normal:
a ) , if reduces : : O ² Á : ž ³
)
b i
c ) c , if is invertible

)
d ² ³ , for any polynomial ²%³  -´%µ

)
2 For any #Á $  = ,
º#Á $» ~ º #Á i i $»

and, in particular,
) ) ) i # ) #~