Hermitian Inner Product

A Hermitian inner product on a complex vector space is a sesquilinear form on taking values in the complex numbers. It is antilinear in the second argument and 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 sesquilinear form

(1)

on , where and . Note that by writing , it is possible to consider , in which case is the usual inner product on Euclidean space 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 a positive definite symmetric bilinear form, and its imaginary part a symplectic form by properties 5 and 6. A matrix defines a sesquilinear 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.