Sesquilinear form: Encyclopedia II - Sesquilinear form - Hermitian form

Sesquilinear form - Hermitian form

A Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V × V → C such that

The standard Hermitian form on Cⁿ is given by

More generally, the inner product on any Hilbert space is a Hermitian form.

If V is a finite-dimensional space, then relative to any basis {e_i} of V, a Hermitian form is represented by a Hermitian matrix H:

The components of H are given by H_ij = h(e_i, e_j).

The quadratic form associated to a Hermitian form

is always real. Actually one can show that a sesquilinear form is Hermitian iff the associated quadratic form is real for all z ∈ V.

Adapted from the Wikipedia article "Hermitian form", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki