U
{ RU ¼
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
1
ffiffi ffi
2
p À
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A
cos θ À sin θ
sin θ
cos θ
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
À
i ffiffi ffi
2
p
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A
¼
e
iθ
0
0 e
Àiθ
:
ð14:96Þ
A trace of the resulting matrix is 2 cos θ. In the case of π < θ < 2π, we get a
diagonal matrix similarly. The conformation is left for readers.
14.5 Hermitian Quadratic Forms
The Hermitian quadratic forms appeared in, e.g., (13.34) and (13.83) in relation to
Gram matrices in Sect. 13.2. The Hermitian quadratic forms have wide applications
in the field of mathematical physics and materials science.
Let H be an Hermitian operator and jxi on an inner vector space. We define the
Hermitian quadratic form in an arbitrary orthonormal basis as follows:
x H
j jx
h
i x
Ã
1 Á Á Áx
Ã
n
À
Á
H
x 1
⋮
x n
0
B
@
1
C
A ¼
X
i,j
x
Ã
i H
ð Þ ij x j ,
where jxi is represented as a column vector, as already mentioned in Sect. 13.4. Let
us start with unitary diagonalization of (13.40), where a Gram matrix is a kind of
Hermitian matrix. Following similar procedures, as in the case of (13.36) we obtain a
diagonal matrix and an inner product such that
x H
j jx
h
i¼ ξ
Ã
1 Á Á Áξ
Ã
n
À
Á
λ 1 Á Á Á 0
⋮ ⋱ ⋮
0 Á Á Á λ n
0
B
@
1
C
A
ξ 1
⋮
ξ n
0
B
@
1
C
A ¼ λ 1 ξ 1 j
2
þ Á Á Á þ λ n
ξ n
2
: ð14:97Þ
Notice, however, that the Gram matrix comprising basis vectors (that are linearly
independent) is positive definite. Remember that if a Gram matrix is constructed by
A
{ A (Hermitian as well) according to whether A is non-singular or singular, A
{ A is
either positive definite or non-negative. The Hermitian matrix we are dealing with
here, in general, does not necessarily possess the positive definiteness or
non-negative feature. Yet, remember that hx| H| xi and eigenvalues λ 1 , Á Á Á, λ n are real.
Positive definiteness of matrices is an important concept in relation to the
Hermitian (and real) quadratic forms (see Sect. 13.2). In particular, in the case
where all the matrix elements are real and | xi is defined in a real domain, we are
568
14 Hermitian Operators and Unitary Operators
{ RU ¼
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
1
ffiffi ffi
2
p À
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A
cos θ À sin θ
sin θ
cos θ
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
À
i ffiffi ffi
2
p
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A
¼
e
iθ
0
0 e
Àiθ
:
ð14:96Þ
A trace of the resulting matrix is 2 cos θ. In the case of π < θ < 2π, we get a
diagonal matrix similarly. The conformation is left for readers.
14.5 Hermitian Quadratic Forms
The Hermitian quadratic forms appeared in, e.g., (13.34) and (13.83) in relation to
Gram matrices in Sect. 13.2. The Hermitian quadratic forms have wide applications
in the field of mathematical physics and materials science.
Let H be an Hermitian operator and jxi on an inner vector space. We define the
Hermitian quadratic form in an arbitrary orthonormal basis as follows:
x H
j jx
h
i x
Ã
1 Á Á Áx
Ã
n
À
Á
H
x 1
⋮
x n
0
B
@
1
C
A ¼
X
i,j
x
Ã
i H
ð Þ ij x j ,
where jxi is represented as a column vector, as already mentioned in Sect. 13.4. Let
us start with unitary diagonalization of (13.40), where a Gram matrix is a kind of
Hermitian matrix. Following similar procedures, as in the case of (13.36) we obtain a
diagonal matrix and an inner product such that
x H
j jx
h
i¼ ξ
Ã
1 Á Á Áξ
Ã
n
À
Á
λ 1 Á Á Á 0
⋮ ⋱ ⋮
0 Á Á Á λ n
0
B
@
1
C
A
ξ 1
⋮
ξ n
0
B
@
1
C
A ¼ λ 1 ξ 1 j
2
þ Á Á Á þ λ n
ξ n
2
: ð14:97Þ
Notice, however, that the Gram matrix comprising basis vectors (that are linearly
independent) is positive definite. Remember that if a Gram matrix is constructed by
A
{ A (Hermitian as well) according to whether A is non-singular or singular, A
{ A is
either positive definite or non-negative. The Hermitian matrix we are dealing with
here, in general, does not necessarily possess the positive definiteness or
non-negative feature. Yet, remember that hx| H| xi and eigenvalues λ 1 , Á Á Á, λ n are real.
Positive definiteness of matrices is an important concept in relation to the
Hermitian (and real) quadratic forms (see Sect. 13.2). In particular, in the case
where all the matrix elements are real and | xi is defined in a real domain, we are
568
14 Hermitian Operators and Unitary Operators