Sesquilinear - Definition and Overview

In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. (The name originates from the Latin root sesqui- meaning "one and a half"). Compare with a bilinear form, which is linear in both arguments.

Conventions differ as to which argument should be antilinear. We take the first to be antilinear and the second to be linear. This is the physicist's convention — originating in Dirac's bra-ket notation in quantum mechanics — but is becoming more popular among mathematicians as well.

Specifically a map φ : V × V → C is sesquilinear if

<math>\phi(x + y, z + w) = \phi(x, z) + \phi(x, w) + \phi(y, z) + \phi(y, w)\,<math>

<math>\phi(a x, y) = \bar{a}\,\phi(x,y)<math>

<math>\phi(x, ay) = a\,\phi(x,y)<math>

for all x,y,z,w ∈ V and all a ∈ C.

For a fixed z in V the map <math>w \mapsto \phi(z,w)<math> is a linear functional on V (i.e. an element of the dual space V*). Likewise, the map <math>w \mapsto \phi(w,z)<math> is an antilinear functional on V.

Given any sesquilinear form φ on V we can define a second sesquilinear form ψ via the conjugate transpose:

<math>\psi(w,z) = \overline{\phi(z,w)}<math>

In general, ψ and φ will be different. If they are the same then φ is said to be Hermitian. If they are negatives of one another, then φ is said to be skew-Hermitian. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.

Hermitian forms

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

<math>h(w,z) = \overline{h(z, w)}<math>

The standard Hermitian form on Cⁿ is given by

<math>\langle w,z \rangle = \sum_{i=1}^n\overline{w}_i z_i<math>

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:

<math>h(w,z) = \overline{\mathbf{w}}^T \mathbf{Hz}<math>

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

The quadratic form assoctiated 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.

Skew-Hermitian forms

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

<math>\varepsilon(w,z) = -\overline{\varepsilon(z, w)}<math>

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:

<math>\varepsilon(w,z) = \overline{\mathbf{w}}^T \mathbf{Az}<math>

The quadratic form assoctiated to a skew-Hermitian form

Q(z) = ε(z,z)

is always pure imaginary.