H ¼ H
{ ,
ð1:119Þ
it is said to be an Hermitian matrix or a self-adjoint matrix. We often say that such a
matrix is Hermitian.
Since the Gram matrices frequently appear in matrix algebra and play a role, their
properties are worth examining. Since G is an Hermitian matrix, it can be diagonalized through similarity transformation using a unitary matrix. We will give its proof
later (see Sect. 14.3).
Let us deal with (13.34) further. We have
xjx
h i ¼ x
Ã
1 Á Á Áx
Ã
n
À
Á
UU
{
e 1 je 1
h
i Á Á Á e 1 je n
h
i
⋮
⋱
⋮
e n je 1
h
i Á Á Á e n je n
h
i
0
B
@
1
C
AUU
{
x 1
⋮
x n
0
B
@
1
C
A,
ð13:36Þ
where U is defined as UU
{
¼ U
{ U ¼ E. Such a matrix U is called a unitary matrix.
We represent a matrix form of U as
U ¼
u 11 Á Á Á u 1n
⋮ ⋱ ⋮
u n1 Á Á Á u nn
0
B
@
1
C
A, U
{
¼
u
Ã
11
Á Á Á u
Ã
n1
⋮ ⋱ ⋮
u
Ã
1n Á Á Á u
Ã
nn
0
B
@
1
C
A:
ð13:37Þ
Here, putting
U
{
x 1
⋮
x n
0
B
@
1
C
A ¼
ξ 1
⋮
ξ n
0
B
@
1
C
A or equivalently
X n
k¼1
x k u
Ã
ki ξ i
ð13:38Þ
and taking its adjoint such that
x
Ã
1 Á Á Áx
Ã
n
À
Á
U ¼ ξ
Ã
1 Á Á Áξ
Ã
n
À
Á
or equivalently
X n
k¼1
x
Ã
k u ki ¼ ξ
Ã
i ,
we have
xjx
h i ¼ ξ
Ã
1 Á Á Áξ
Ã
n
À
Á
U
{
e 1 je 1
h
i Á Á Á e 1 je n
h
i
⋮
⋱
⋮
e n je 1
h
i Á Á Á e n je n
h
i
0
B
@
1
C
AU
ξ 1
⋮
ξ n
0
B
@
1
C
A:
ð13:39Þ
We assume that the Gram matrix is diagonalized by a similarity transformation by
U. After being diagonalized, similarly to (12.192) and (12.193) the Gram matrix has
a following form G
0 :
530
13 Inner Product Space