Page 261 - Advanced Linear Algebra
P. 261
Structure Theory for Normal Operators 245
)
1 is unitary/orthogonal
)
2 is a product of reflections.
Proof. Since reflections are unitary/orthogonal and the product of unitary/
orthogonal operators is unitary, it follows that 2) implies 1). For the converse,
=
8
let be unitary. Let ~²" Á Ã Á " ³ be an orthonormal basis for . Then
/ ² "c" " ³ ~ "
and so if %~ " c " then
²/ % ³" ~ "
that is, % / is the identity on º" » . Suppose that we have found
reflections / Á % Ã Á / % c for which c / c Ä % / % is the identity on
». Then
º" ÁÃÁ" c
/ ² c "c" " ³ ~ c "
Moreover, we claim that ² " c c " ³ " for , since
º c " c " Á " » ~ º²/ % c Ä/ % ³" Á " »
%
~º " Á / Ä/ % c " »
~º " Á " »
~º" Á " »
~
Hence, if %~ c " c " , then
%
²/ Ä/ % % ³" ~ / " ~ "
and so % / Ä/ % is the identity on º" ÁÃÁ" » . Thus, for ~ we
~ % , as desired.
have /Ä/ % % and so ~ / Ä/ %
The Structure of Normal Operators
The following theorem includes the spectral theorems stated above for real and
complex normal operators, along with some further refinements related to self-
adjoint and unitary/orthogonal operators.
Theorem 10.18 (The structure theorem for normal operators )
1 )(Complex case ) Let = be a finite-dimensional complex inner product
space.
a The following are equivalent for ²= ³ :
)
B
i) is normal
)
ii is unitarily diagonalizable
)
iii has an orthogonal spectral resolution
~ b Ä
b
