As expected, we have a diagonal matrix, one of eigenvalues for which is zero.
That is, the Gram matrix is non-negative.
In the present case, let us think of a following Hermitian quadratic form:
ϕ x 1 , x 2
ð
Þ¼
X 2
i,j¼1
x
Ã
i G
ð Þ ij x j ¼ x
Ã
1 x
Ã
2
À
Á
UU
{ GUU
{ x 1
x 2
¼ x
Ã
1 x
Ã
2
À
Á
U
2 0
0 0
U
{ x 1
x 2
¼ ξ
Ã
1 ξ
Ã
2
À
Á 2 0
0 0
ξ 1
ξ 2
¼ 2 ξ 1
j j
2 :
Then, if we take
ξ 1
ξ 2
¼
0
ξ 2
with ξ 2 6 ¼ 0 as a column vector, we get ϕ(x 1 ,
x 2 ) ¼ 0. With this type of column vector, we have
U
{ x 1
x 2
¼
0
ξ 2
, i:e:,
x 1
x 2
¼ U
0
ξ 2
¼
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
À
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A
0
ξ 2
¼
1
ffiffi ffi
2
p
ξ 2
ξ 2
,
where ξ 2 is an arbitrary complex number. Thus, even for
x 1
x 2
6 ¼
0
0
we may
have ϕ(x 1 , x 2 ) ¼ 0. To be more specific, if we take, e.g.,
x 1
x 2
¼
1
1
or
1
2
, we
get
ϕ 1, 1
ð Þ ¼ 1 1
ð
Þ
1 À1
À1 1
1
1
¼ 1 1
ð
Þ
0
0
¼ 0,
ϕ 1, 2
ð Þ ¼ 1 2
ð
Þ
1 À1
À1 1
1
2
¼ 1 2
ð
Þ
À1
1
¼ 1 > 0:
13.3 Adjoint Operators
A linear transformation A is similarly defined as before and A transforms jxi of
(13.32) such that
13.3 Adjoint Operators
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