Sesquilinear Form - Skew-Hermitian Form

A skew-Hermitian form (also called an antisymmetric sesquilinear form), is a sesquilinear form ε : V × V → C such that

Every skew-Hermitian form can be written as i times a Hermitian form.

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

The quadratic form associated to a skew-Hermitian form

Q(z) = ε(z,z)

is always pure imaginary.