Sesquilinear form#Hermitian form

As Hermitian product, Hermitian sesquilinear Hermitian form or simply ( after Charles Hermite ) is called in linear algebra a special kind of sesquilinear similar to the symmetric bilinear forms.

Definition

Be a vector space over the field. A Hermitian sesquilinear form is a mapping

Which satisfies for all and for all of the following conditions:

It refers to complex conjugation.

For the sequence of linear and semilinear argument, there are different conventions.

With the property (3) follows an (1) from (2) and (2) from (1). For clarity, here are but both (1 ) and ( 2) as the prerequisites.

Relevant to the concept of Hermitian sesquilinear is only over the field of complex numbers; over the field of real numbers is any Hermitian sesquilinear form a symmetric bilinear form. The inner product on a complex vector space is a Hermitian sesquilinear. Similarly, also referred to a sesquilinear form on any module as a Hermitian if and only if for any involutive Antiautomorphismus on the module underlying ring. Located in the center of the ring, it means the sesquilinear - Hermitian if and only if the following holds.

Hermitian standard form

The through

Defined mapping is called Hermitian standard form.