Page 253 - Advanced Linear Algebra

Structure Theory for Normal Operators 237

)
3 has an orthogonal spectral resolution
~ b Ä (10.1)
b
where bÄb and ~ is orthogonal for all , in which case,

¸Á à Á ¹ is the spectrum of and
ker
im²³ ~ ; and ²³ ~ ;

£
Proof. We have seen that 1) and 2) are equivalent. To see that 2) and 3) are
equivalent, Theorem 8.12 says that
=~ ; l Ä l ;

if and only if
~ b Ä
b
and in this case,

ker

im²³ ~ ; and ²³ ~ ;
£
; ; ž for £ if and only if
But

im²³ ž ker ²³

that is, if and only if each is orthogonal. Hence, the direct sum =~

; lÄl ; is an orthogonal sum if and only if each projection is

orthogonal.
The Real Case
If -~ s , then ²%³ has the form

²%³ ~ ²% c ³Ä²% c ³ ²%³Ä ²%³

where each ²%³ is an irreducible monic quadratic. Hence, the primary cyclic

gives
decomposition of =
=~ ; p Ä p ; p > p Ä p >
where > is cyclic with prime quadratic order ² % ³ . Therefore, Theorem 8.8
implies that there is an ordered basis for which
8
c
´O µ ~ > > 8 ?

Theorem 10.10 (The spectral theorem for normal operators: real case ) A
linear operator on a finite-dimensional real inner product space is normal if

and only if