ajβb þ γc
h
i¼ β ajb
h iþ γ ajc
h i:
ð13:18Þ
Also from (13.2),
βb þ γcja
h
i¼ ajβb þ γc
h
i
à ¼ β ajb
h iþ γ ajc
h i
½
à ¼ β
à bja
h iþ γ
à cja
h i: ð13:19Þ
That is,
hβb þ γcj ¼ β
Ã
hb j þγ
Ã
hcj :
ð13:20Þ
Therefore, when we take out a scalar from a bra vector, it should be a complex
conjugate. When the scalar is taken out from a ket vector, on the other hand, it is
unaffected by definition (13.3). To show this, we have jαai ¼ αjai. Taking its
adjoint, hαaj ¼ α
Ã
haj.
We can view (13.17) as a linear transformation in a vector space. In other words,
if we regard j ∙ i as a linear transformation of a vector a 2 V
n to j ai 2 f
V
n , j ∙ i is that
of V
n to f
V
n . On the other hand, h ∙ j could not be regarded as a linear transformation
of a 2 V
n to haj2 f
V
n
0
. Sometimes the said transformation is referred to as “antilinear”
or “sesquilinear.” From the point of view of formalism, the inner product can be
viewed as an operation: f
V
n
 f
V
n
0 ! ℂ.
Let us consider jxi ¼ jyi in an inner product space f
V
n
. Then jxi Àjyi ¼ 0. That is,
jx À yi ¼ 0. Therefore, we have x ¼ y, or j0i ¼ 0. This means that the linear
transformation j ∙ i converts 0 2 V
n to j 0i 2 f
V
n . This is a characteristic of a linear
transformation represented in (11.44). Similarly we have h0j ¼ 0. However, we do
not have to get into further details in this book. Also if we have to specify a vector
space, we simply do so by designating it as V
n .
13.2 Gram Matrices
Once we have defined an inner product between any pair of vectors jai and jbi of a
vector space V
n , we can define and calculate various quantities related to inner
products. As an example, let ja 1 i, Á Á Á , ja n i and jb 1 i, Á Á Á , jb n i be two sets of vectors
in V
n . The vectors ja 1 i, Á Á Á , ja n i may or may not be linearly independent. This is true
of jb 1 i, Á Á Á , jb n i. Let us think of a following matrix M defined as below:
M ¼
ha 1 j
⋮
ha n j
0
B
@
1
C
A jb 1 iÁ Á Ájb n i
ð
Þ¼
a 1 jb 1
h
i Á Á Á a 1 jb n
h
i
⋮
⋱
⋮
a n jb 1
h
i Á Á Á a n jb n
h
i
0
B
@
1
C
A:
ð13:21Þ
We assume following cases.
526
13 Inner Product Space
h
i¼ β ajb
h iþ γ ajc
h i:
ð13:18Þ
Also from (13.2),
βb þ γcja
h
i¼ ajβb þ γc
h
i
à ¼ β ajb
h iþ γ ajc
h i
½
à ¼ β
à bja
h iþ γ
à cja
h i: ð13:19Þ
That is,
hβb þ γcj ¼ β
Ã
hb j þγ
Ã
hcj :
ð13:20Þ
Therefore, when we take out a scalar from a bra vector, it should be a complex
conjugate. When the scalar is taken out from a ket vector, on the other hand, it is
unaffected by definition (13.3). To show this, we have jαai ¼ αjai. Taking its
adjoint, hαaj ¼ α
Ã
haj.
We can view (13.17) as a linear transformation in a vector space. In other words,
if we regard j ∙ i as a linear transformation of a vector a 2 V
n to j ai 2 f
V
n , j ∙ i is that
of V
n to f
V
n . On the other hand, h ∙ j could not be regarded as a linear transformation
of a 2 V
n to haj2 f
V
n
0
. Sometimes the said transformation is referred to as “antilinear”
or “sesquilinear.” From the point of view of formalism, the inner product can be
viewed as an operation: f
V
n
 f
V
n
0 ! ℂ.
Let us consider jxi ¼ jyi in an inner product space f
V
n
. Then jxi Àjyi ¼ 0. That is,
jx À yi ¼ 0. Therefore, we have x ¼ y, or j0i ¼ 0. This means that the linear
transformation j ∙ i converts 0 2 V
n to j 0i 2 f
V
n . This is a characteristic of a linear
transformation represented in (11.44). Similarly we have h0j ¼ 0. However, we do
not have to get into further details in this book. Also if we have to specify a vector
space, we simply do so by designating it as V
n .
13.2 Gram Matrices
Once we have defined an inner product between any pair of vectors jai and jbi of a
vector space V
n , we can define and calculate various quantities related to inner
products. As an example, let ja 1 i, Á Á Á , ja n i and jb 1 i, Á Á Á , jb n i be two sets of vectors
in V
n . The vectors ja 1 i, Á Á Á , ja n i may or may not be linearly independent. This is true
of jb 1 i, Á Á Á , jb n i. Let us think of a following matrix M defined as below:
M ¼
ha 1 j
⋮
ha n j
0
B
@
1
C
A jb 1 iÁ Á Ájb n i
ð
Þ¼
a 1 jb 1
h
i Á Á Á a 1 jb n
h
i
⋮
⋱
⋮
a n jb 1
h
i Á Á Á a n jb n
h
i
0
B
@
1
C
A:
ð13:21Þ
We assume following cases.
526
13 Inner Product Space