Hermitian Inner Product

A Hermitian inner product on a complex vector space is a complex-valued bilinear form on which is antilinear in the second slot, and is positive definite. That is, it satisfies the following properties, where denotes the complex conjugate of .

1.

2.

3.

4.

5.

6. , with equality only if

The basic example is the form

(1)

on , where and . Note that by writing , it is possible to consider , in which case is the Euclidean inner product and is a nondegenerate alternating bilinear form, i.e., a symplectic form. Explicitly, in , the standard Hermitian form is expressed below.

(2)

A generic Hermitian inner product has its real part symmetric positive definite, and its imaginary part symplectic by properties 5 and 6. A matrix defines an antilinear form, satisfying 1-5, by iff is a Hermitian matrix. It is positive definite (satisfying 6) when is a positive definite matrix. In matrix form,

(3)

and the canonical Hermitian inner product is when is the identity matrix.