Sesquilinear Form - Hermitian Form

The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain differential form on a Hermitian manifold.

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 complex Hilbert space is a Hermitian form.

A vector space with a Hermitian form (V,h) is called a Hermitian space.

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

Q(z) = h(z,z)

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