1.7 Inner Product Spaces
13
[T ⊗ S](|v ⊗ |w) = [T ⊗ S]
n
i=1
a i |v i ⊗ |w i
≡
n
i=1
a i T |v i ⊗ S|w i ,
(where we have written T |v i meaning T (|v i )) is a linear operator on V ⊗ W .
More generally, assume that R : V 1 −→ V 2 and S : W 1 −→ W 2 are two linear
transformations, where the vector spaces are distinct. Then any linear transformation
T : V 1 ⊗ W 1 −→ V 2 ⊗ W 2 can be represented by linear combinations of tensor products of linear transformations R i : V 1 −→ V 2 and S i : W 1 −→ W 2 , i = 1, 2, . . . n,
i.e., T =
n
i=1
z i R i ⊗ S i , where z i ∈ C. By definition, we have
T (|v ⊗ |w) =
n
i=1
z i R i ⊗ S i
(|v ⊗ |w) =
n
i=1
(z i R i (|v)) ⊗ S i (|w).
Definition 1.7.11 Let V and W be two vector spaces (over the same field K )
endowed with inner products, and consider the tensor product V ⊗ W . Then, an
inner product (·, ·) V ⊗W : (V ⊗ W ) × (V ⊗ W ) −→ K can be defined naturally by
⎛
⎝
n
i=1
a i |v i ⊗ |w i ,
m
j=1
a i |v
j ⊗ |w
j
⎞
⎠ ≡
n
i=1
m
j=1
a
∗
i b j v i |v
j w i |w
j .
In the following we define the outer product representation.
Definition 1.7.12 Assume that V and W are two inner product spaces. For every
|v ∈ V and |w ∈ W , we define the linear transformation |wv| : V −→ W by
(|wv|)(|v
) = =v|v
|w.
In this context, it is possible to define the concept of projector onto a subspace.
Definition 1.7.13 Suppose that {|1, |2, . . . , |n} is an orthonormal basis for the
n-dimensional inner product space V such that {|1, |2, . . . , |m} is an orthonormal
basis for a m-dimensional subspace W of V . Then we can define the projector
P ≡
m
i=1
|ii| onto the subspace W .
Exercise 1.7.2 Show that P ≡
m
i=1
|ii| given in Definition 1.7.13 does not depend
on the choice of the orthogonal basis for W .
Fortunately, the characteristic of certain types of operators are maintained by
applying tensor products V ⊗ W of vectors spaces.
Proposition 1.7.2 Let T 1 : V −→ V and T 2 : W −→ W be linear operators.
13
[T ⊗ S](|v ⊗ |w) = [T ⊗ S]
n
i=1
a i |v i ⊗ |w i
≡
n
i=1
a i T |v i ⊗ S|w i ,
(where we have written T |v i meaning T (|v i )) is a linear operator on V ⊗ W .
More generally, assume that R : V 1 −→ V 2 and S : W 1 −→ W 2 are two linear
transformations, where the vector spaces are distinct. Then any linear transformation
T : V 1 ⊗ W 1 −→ V 2 ⊗ W 2 can be represented by linear combinations of tensor products of linear transformations R i : V 1 −→ V 2 and S i : W 1 −→ W 2 , i = 1, 2, . . . n,
i.e., T =
n
i=1
z i R i ⊗ S i , where z i ∈ C. By definition, we have
T (|v ⊗ |w) =
n
i=1
z i R i ⊗ S i
(|v ⊗ |w) =
n
i=1
(z i R i (|v)) ⊗ S i (|w).
Definition 1.7.11 Let V and W be two vector spaces (over the same field K )
endowed with inner products, and consider the tensor product V ⊗ W . Then, an
inner product (·, ·) V ⊗W : (V ⊗ W ) × (V ⊗ W ) −→ K can be defined naturally by
⎛
⎝
n
i=1
a i |v i ⊗ |w i ,
m
j=1
a i |v
j ⊗ |w
j
⎞
⎠ ≡
n
i=1
m
j=1
a
∗
i b j v i |v
j w i |w
j .
In the following we define the outer product representation.
Definition 1.7.12 Assume that V and W are two inner product spaces. For every
|v ∈ V and |w ∈ W , we define the linear transformation |wv| : V −→ W by
(|wv|)(|v
) = =v|v
|w.
In this context, it is possible to define the concept of projector onto a subspace.
Definition 1.7.13 Suppose that {|1, |2, . . . , |n} is an orthonormal basis for the
n-dimensional inner product space V such that {|1, |2, . . . , |m} is an orthonormal
basis for a m-dimensional subspace W of V . Then we can define the projector
P ≡
m
i=1
|ii| onto the subspace W .
Exercise 1.7.2 Show that P ≡
m
i=1
|ii| given in Definition 1.7.13 does not depend
on the choice of the orthogonal basis for W .
Fortunately, the characteristic of certain types of operators are maintained by
applying tensor products V ⊗ W of vectors spaces.
Proposition 1.7.2 Let T 1 : V −→ V and T 2 : W −→ W be linear operators.