Page 264 - Advanced Linear Algebra

248 Advanced Linear Algebra

arbitrary set of of its values at the distinct points Á Ã Á . This follows from
the Lagrange interpolation formula
c v %c y
²%³ ~ ² ³
~ w £ c z
Therefore, we can define a unique polynomial ²%³ by specifying the values
² ³ , for ~ ÁÃÁ .

For example, for a given   , if ²%³ is a polynomial for which

² ³ ~ Á

for ~ Á Ã Á , then
² ³ ~

and so each projection is a polynomial function of . As another example, if

is invertible and ² ³ ~ c , then

² ³ ~ c b Ä b c ~ c

, then since
as can easily be verified by direct calculation. Finally, if ² ³ ~
each is self-adjoint, we have

² ³ ~ b Ä b ~ i

and so is a polynomial in .
i

We can extend this idea further by defining , for any function
¢ ¸ Á Ã Á ¹ ¦ -
the linear operator ² ³ by

² ³ ~ ² ³ b Ä b ² ³

Á
For example, we may define j c Á and so on. Notice, however, that
since the spectral resolution of is a finite sum, we gain nothing (but

convenience) by using functions other than polynomials, for we can always find

a polynomial ²%³ for which ² ³ ~ ² ³ for ~ Á Ã Á and so
² ³ ~ ² ³. The study of the properties of functions of an operator is

referred to as the functional calculus of .

=
According to the spectral theorem, if is complex and is normal, then ² ³ is
=

a normal operator whose eigenvalues are ² ³ . Similarly, if is real and is
symmetric, then ² ³ is symmetric, with eigenvalues ² ³ .