Sesquilinear form

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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 numerical prefix sesqui- meaning "one and a half". Compare with a bilinear form, which is linear in both arguments; although many authors, especially when working solely in a complex setting, refer to sesquilinear forms as bilinear forms.

A motivating example is the inner product on a complex vector space, which is not bilinear, but instead sesquilinear. See geometric motivation below.

1 Definition and conventions
2 Geometric motivation
3 Hermitian form
4 Skew-Hermitian form
5 Generalization: over a *-ring
6 References

Conventions differ as to which argument should be linear. We take the first to be conjugate-linear and the second to be linear. This is the convention used by essentially all physicists and originates in Dirac's bra-ket notation in quantum mechanics. The opposite convention is perhaps more common in mathematics but is not universal.

Specifically a map $\phi : V \times V \to \mathbb C$ is sesquilinear if

$\begin{align} &\phi(x + y, z + w) = \phi(x, z) + \phi(x, w) + \phi(y, z) + \phi(y, w)\\ &\phi(a x, b y) = \bar a b\,\phi(x,y)\end{align}$

for all x,y,z,w ∈ V and all a, b ∈ C.

A sesquilinear form can also be viewed as a bilinear map

$\bar V \times V \to \mathbb C$

where $\bar V$ is the complex conjugate vector space to V. By the universal property of tensor products these are in one-to-one correspondence with (complex) linear maps

$\bar V \otimes V \to \mathbb C.$

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

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

$\psi(w,z) = \overline{\phi(z,w)}$

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.

Bilinear forms are to squaring ( $z 2$ ), what sesquilinear forms are to Euclidean norm ( $| z | 2 = z * z$ ).

The norm associated to a sesquilinear form is invariant under multiplication by the complex circle (complex numbers of unit norm), while the norm associated to a bilinear form is equivariant (with respect to squaring). Bilinear forms are algebraically more natural, while sesquilinear forms are geometrically more natural.

If B is a bilinear form on a complex vector space and $| x | B : = B (x, x)$ is the associated norm, then $| i x | B = B (i x, i x) = i 2 B (x, x) = - | x | B$ .

By contrast, if S is a sesquilinear form on a complex vector space and $| x | S : = S (x, x)$ is the associated norm, then $|ix|_S = S(ix,ix)=\bar i i S(x,x) = |x|_S$ .

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

$h(w,z) = \overline{h(z, w)}$

The standard Hermitian form on Cⁿ is given by

$\langle w,z \rangle = \sum_{i=1}^n \overline{w_i} z_i.$

More generally, the inner product on any 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:

$h(w,z) = \overline{\mathbf{w}^T} \mathbf{Hz}$

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.

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

$\varepsilon(w,z) = -\overline{\varepsilon(z, w)}$

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:

$\varepsilon(w,z) = \overline{\mathbf{w}}^T \mathbf{Az}$

The quadratic form associated to a skew-Hermitian form

Q(z) = ε(z,z)

is always pure imaginary.

A sesquilinear form and a Hermitian form can be defined over any *-ring, and the examples of symmetric bilinear forms, skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms, are all Hermitian forms for various involutions.

Particularly in L-theory, one also sees the term ε-symmetric form, where $\epsilon=\pm 1$ , to refer to both symmetric and skew-symmetric forms.