Page 247 - Advanced Linear Algebra

Structure Theory for Normal Operators 231

i
i

²´ µ Á98 ³ Á ~º Á » ~º Á » ~º Á » ~²´ µ Á8 9 ³ Á

i are conjugate transposes. The conjugate transpose of a
and so ´µ Á and ´ µ Á 98 8 9
matrix (~ ² ³ is
Á
i
(~ ² ³ !
Á
and is called the adjoint of .
(
=

B
Theorem 10.4 Let ²= Á > ³ , where and > are finite-dimensional inner
product spaces.
)
i
1 The operator adjoint d and the Hilbert space adjoint are related by

d = c k ~²9 ³ i k 9 >
where 9 = and 9 > are the conjugate Riesz isomorphisms on and > = ,
respectively.
)
2 If and are ordered orthonormal bases for and > , respectively, then
=
9
8
i ³ i
´µ Á ~ ²´ µ Á 98 8 9
In words, the matrix of the adjoint is the adjoint conjugate transpose of
(
i
)

the matrix of .
Orthogonal Projections
In an inner product space, we can single out some special projection operators.
is said to be orthogonal .
Definition A projection of the form :Á: ž
Equivalently, a projection is orthogonal if ker²³ ž im ²³ .

Some care must be taken to avoid confusion between orthogonal projections and
two projections that are orthogonal to each other, that is, for which
~ ~ .
We have seen that an operator is a projection operator if and only if it is

idempotent. Here is the analogous characterization of orthogonal projections.
Theorem 10.5 Let be a finite-dimensional inner product space. The following
=
are equivalent for an operator on :
=

)
1 is an orthogonal projection
)
2 is idempotent and self-adjoint
)
3 is idempotent and does not expand lengths, that is
)
) ) ## )
for all #= .