DIAGONALIZATION OF HERMITIAN MATRICES
25
“
U
= [
u
1
,
u
2
,
. . .
,
u
N
] is unitary”.
In the real case, the same equivalence is also true of
orthogonal matrices.
Properties:
1. If
U
is unitary, then
U
†
U
=
I
implies
U
x
,
U
y
=
U
†
(
U
x
),
y
=
x
,
y
for all
x
,
y
∈
V
. This says that when a unitary
U
acts as a linear operator on the
inner product space, all inner products are left unchanged. In particular, all vector
norms and orthogonality relationships remain unchanged.
2. A similar property holds for orthogonal operators
O
on real vector spaces:
O
x
,
O
y
=
x
,
y
for all
x
,
y
∈
V
.
3. When
A
is Hermitian,
x
,
A
y
=
A
x
,
y
(1.15)
for all
x
,
y
∈
V
.
1.10
Diagonalization of Hermitian Matrices
Consider a linear operator
H
that is Hermitian with respect to an inner product. Let
u
1
,
u
2
be two eigenvectors of
H
, with eigenvalues
λ
1
,
λ
2
respectively.
Consider the following
argument:
(
λ
2

¯
λ
1
)
u
1
,
u
2
=
u
1
,
λ
2
u
2

λ
1
u
1
,
u
2
=
u
1
,
H
u
2
 H
u
1
,
u
2
=
u
1
,
H
u
2

u
1
,
H
u
2
= 0
Two important conclusions:
The identity (
λ
2

¯
λ
1
)
u
1
,
u
2
= 0 implies
1. If
u
1
=
u
2
(nonzero so
u
1
,
u
2
=

u
1

2
>
0), then 0 =
λ
2

¯
λ
1
=
λ
1

¯
λ
1
. In other
words, all eigenvalues of a Hermitian operator
H
must be
real
.
2. If
λ
1
=
λ
2
, we must have
u
1
,
u
2
= 0. In other words, two eigenvectors with distinct
eigenvalues of a Hermitian operator
H
must be
orthogonal
.
When
V
=
C
N
and the eigenvalues of a Hermitian matrix
H
are all single roots of
the characteristic polynomial (so there are exactly
n
distinct eigenvalues, all of which are
real), it follows that there are
N
mutually orthogonal eigenvectors of
H
, which of course
form a basis. If these are normalized, we get an
orthonormal basis of eigenvectors
{
u
i
}
N
i
=1
for
V
. With more effort, one can show this result is true for any Hermitian matrix
H
.
An orthonormal basis of eigenvectors corresponds to a matrix of eigenvectors
U
=
[
u
1
,
u
2
,
. . .
,
u
N
] which is
unitary
,
U

1
=
U
†
. Therefore,
H
will be unitarily diagonalizable:
H
=
UDU
†