Page 256 - Advanced Linear Algebra

240 Advanced Linear Algebra

~º ²% b &³Á % b &»

~ º %Á %» b º &Á &» b º %Á &» b º &Á %»

~ º %Á &» b º &Á %»

~ º %Á &» b º%Á &»

~ º %Á &» b º %Á &»

~ º %Á &»

and so ~ .

For part 4), if is Hermitian (-~ d ) and # ~ # , then
#~ #~ i #~ #

and so ~ is real. If is symmetric ( - s ~ ) , we must be a bit careful, since
a nonreal root of ²%³ is not an eigenvalue of . However, matrix techniques

can come to the rescue here. If (~ ´ µ E for any ordered orthonormal basis E
for = , then ² % ³ ~ ( ² % ³ . Now, is a real symmetric matrix, but can be
(
thought of as a complex Hermitian matrix with real entries. As such, it
represents a Hermitian linear operator on the complex space d and so, by what
we have just shown, all (complex) roots of its characteristic polynomial are real.
But the characteristic polynomial of is the same, whether we think of as a
(
(
real or a complex matrix and so the result follows.
Unitary Operators and Isometries
We now turn to the basic properties of unitary operators. These are the
workhorse operators, in that a unitary operator is precisely a normal operator
that maps orthonormal bases to orthonormal bases.
Note that is unitary if and only if

º#Á $» ~ º#Á c $»
for all #Á $  = .
Theorem 10.12 Let = be a finite-dimensional inner product space and let
Á B²= ³.
)
1 If and are unitary/orthogonal, then so are the following:

a ) , for  Á ~d ((
)
b
c ) c , if is invertible.

)
2 is unitary/orthogonal if and only it is an isometric isomorphism.
)
3 is unitary/orthogonal if and only if it takes some orthonormal basis to an
orthonormal basis, in which case it takes all orthonormal bases to
orthonormal bases.
)
4 If is unitary/orthogonal, then the eigenvalues of have absolute value .