Page 266 - Advanced Linear Algebra

250 Advanced Linear Algebra

Then Theorem 10.20 implies that ~ . Hence,

~² b Ä b ³ ² b Ä b ³

~² b Ä b ³² b Ä b ³

~

Á
It follows that for any polynomial ²%Á &³ in two variables,
² Á ³ ~ ² Á ³

Á

So if we choose ²%Á &³ with the property that Á ~ ² Á ³ are distinct, then
² Á ³ ~

Á
Á

and we can also choose ²%³ and ²%³ so that ² Á ³ ~ for all and

² Á ³ ~ for all . Then
² ² Á ³³ ~ ² Á ³ ~

Á Á
~ 8 9 8 9 ~ ~

and similarly, ² ² Á ³³ ~ .
Positive Operators
One of the most important cases of the functional calculus is ²%³ ~ j . %
Recall that the quadratic form associated with a linear operator is

8 ²#³ ~ º #Á #»

Definition A self-adjoint linear operator ²= ³ is
B

)
1 positive if 8²#³ ‚ for all #  =

2 positive definite if 8²#³ € for all # £ .
)

Theorem 10.22 A self-adjoint operator on a finite-dimensional inner product

space is
1 positive if and only if all of its eigenvalues are nonnegative
)
2 positive definite if and only if all of its eigenvalues are positive.
)

Proof. If 8²#³ ‚ and # ~ # , then

 º #Á#» ~ º#Á#»