The complex scalar product defined in the previous section is a specific example of a sesquilinear, conjugate-symmetric pairing. We consider these properties in sequence.

A sesquilinear pairing on is a map which satisfies property (HIP2), namely it is linear in the first variable and conjugate linear in the second1 . A conjugate-symmetric sesquilinear pairing is a sesquilinear pairing that also satisfies (HIP1). These pairings admit a matrix representation as in the real case discussed above. Again, we assume we are looking at coordinate vectors with respect to the standard basis for .

Before stating the result, we need to record

Proof
The proof is essentially the same as in the real case, with some minor modifications. is uniquely characterized by its values on ordered pairs of basis vectors; moreover two bilinear pairings are equal precisely if for all pairs . So define be the matrix with entry given by By construction, the pairing is sesquilinear, and agrees with on ordered pairs of basis vectors. Thus the two agree everywhere. This establishes a 1-1 correspondence (sesquilinear pairings on ) ( complex matrices). By construction, the matrix will be conjugate-symmetric iff , or equivalently if is Hermitian. Thus this correspondence restricts to a 1-1 correspondence (conjugate-symmetric sesquilinear pairings on ) ( Hermitian matrices).

In other words, it also satisfies property (HIP3). For this reason we call a Hermitian matrix positive definite iff all of its eigenvalues (which are real numbers) are positive.

It is important to know when a sesquilinear conjugate-symmetric (complex) pairing - which in the real case reduces to a symmetric bilinear pairing - actually represents an inner product; in other words, is positive definite. The following theorem does that for us.

Proof
The proof is essentially the same as in the real case. Let be an orthonormal basis for consisting of eigenvectors of , with (this basis exists as a consequence of Shur’s Theorem and its corollaries). If , then It is easily seen that this sum satisfies condition (HIP3) iff .

Again as in the real case, we can extend this result to arbitrary vector spaces over .

Write down a proof of this theorem, using the proofs of the previous two theorems as a guide.