Let be a vector space over or .
An inner product on is a function such that for all and all , the following properties hold:
Symmetry: ,
Bilinearity: and ,
Nondegeneracy: If for all , then .
This is similar to Definition 2.2.1, except that we have replaced the conjugate symmetry and sesquilinearity with regular symmetry and bilinearity.
We also forego the positive-definiteness condition, as it is often too restrictive in many physical applications.
With Dirac's bra-ket notation, we can write the inner product as , with the subscript to distinguish it from the usual sesquilinear inner product .
Let be an endomorphism on a vector space .
Then the adjoint of , denoted by , is the unique endomorphism satisfying
An operator is said to be self-adjoint if .
It is said to be skew if .
It is easy to see that for all .
To see this, we have
and since is nondegenerate, we must have .
Proposition 2.4.4
An endomorphism on a vector space is skew if and only if for all .
Proof.
() Assume that is skew.
Then for all , we have
which implies that .
() Assume that for all .
Then for all and all , we have
Expanding this out using bilinearity, we get
Since and by assumption, we have
Since this holds for all , we must have
As is nondegenerate, we must have , i.e., is skew.
Notice the following. Recall from Theorem 2.3.8 that in a sesquilinear inner product space, we have if and only if for all .
Here, in a bilinear inner product space, we have is skew if and only if for all .
This demonstrates how much positive-definiteness restricts the structure of an inner product space.
A complex structure on a real vector space is an endomorphism such that and for all (i.e., is an isometry).
We dropped the subscript in the inner product here since is a real vector space, and is just the usual inner product .
The reason they are important is that they eventually allow us to relate real vector spaces to complex vector spaces.
This is important; many physical systems are naturally described using real vector spaces, but complex vector spaces often have nicer mathematical properties.
For instance, eigenvalues may be available only in the complex field.
If you have tried to diagonalize a matrix or find its Jordan normal form, using complex numbers often makes the process much easier.
Proposition 2.4.6
All complex structures on a real inner product space are skew.
Proof.
Let be a complex structure on a real inner product space .
Let be arbitrary, and let .
Then, we have
But as the inner product for real vector spaces is symmetric, we have .
Thus, we have
which implies that for all .
By Proposition 2.4.4, we conclude that is skew.
Given a real inner product space , a complex structure on exists only if is even.
In particular, the set forms an orthonormal basis for , where .
Proof. We can prove this by construction using a Gram-Schmidt-like process.
Start with any nonzero vector .
Normalize it to get .
Then, define .
As is an isometry, we have .
Moreover, as is skew by Proposition 2.4.6, we have
due to Proposition 2.4.4.
Thus, is an orthonormal set.
Now, consider the orthonormal subset .
For each , define .
Therefore, there will always be an even number of vectors in the orthonormal subset.
If any new vector is linearly independent from the existing set, we simply add both the vector and its image under to the orthonormal set.
Continuing this process, we eventually obtain an orthonormal basis for of the desired form.
So what has all this built up to?
Complex structures, as we have seen, allow us to take an orthonormal basis of a real vector space and pair up the basis vectors, effectively doubling the dimension.
This should remind you of how complex numbers can be represented as pairs of real numbers.
By introducing to the set of real numbers, we double the dimension of the space.
Similarly, complex structures allow us to introduce a notion of "imaginary" directions in a real vector space, effectively turning it into a complex vector space.
This process is called complexification and is defined below.
Let be a real vector space.
Then, the complexification of , denoted by , is defined as the tensor product
with the scalar multiplication defined by
for all , , and .
The complexification can be thought of as the set of all formal linear combinations with and .
In other words, the basis vectors of are of the form and , where is a basis for .
The dimensions scale as follows:
( denotes the dimension of as a complex vector space, while denotes its dimension as a real vector space.)
The inner product on is defined as a sesquilinear (Hermitian) inner product with
The complex structure acts similar to multiplication by .
In particular, a dimension- real vector space with a complex structure can be identified with a dimension- complex vector space, where the tensor product is replaced by the action of :
If the basis of is , then the corresponding basis of is .