Sesquilinear form

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.

Definition and conventions

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 φ : V × V → 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 complex bilinear map

$\bar V \times V \to \mathbf{C}$

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

$\bar V \otimes V \to \mathbf{C}.$

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:

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

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.

Geometric motivation

Bilinear forms are to squaring (z²), 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 $|ix|_B = B(ix,ix) = i^{2}B(x,x) = -|x|_B$.

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

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

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

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 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:

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

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.

Skew-Hermitian form

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.

Generalization: over a *-ring

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.

References

• Hazewinkel, Michiel, ed. (2001), "Sesquilinear form", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=Sesquilinear_form&oldid=13338