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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 Latin numerical prefix sesqui- meaning "one and a half". Compare with a bilinear form, which is linear in both arguments. However 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 (i.e. antilinear) 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 more common in mathematicsPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park..

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

$\begin{array}{rl}& \varphi (x+y,z+w)=\varphi (x,z)+\varphi (x,w)+\varphi (y,z)+\varphi (y,w)\\ & \varphi (ax,by)=\overline{a}b\varphi (x,y)\end{array}$

for all x,y,z,w ∈ V and all a, b ∈ C. $\overline{a}$ is the complex conjugate of a.

A sesquilinear form can also be viewed as a complex bilinear map

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

where $\overline{V}$ 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

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

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

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 $|ix{|}_S=S(ix,ix)=\overline{i}iS(x,x)=|x{|}_S$.

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 (using again the "physics" convention of linearity in the second and conjugate linearity in the first variable) by

$⟨w,z⟩=\sum _{i=1}^n\overline{w_i}{z}_i.$

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:

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

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

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

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:

$\epsilon (w,z)=\stackrel{T}{\stackrel{‾}{\mathbf{w}}}\mathbf{A}\mathbf{z}.$

The quadratic form associated to a skew-Hermitian form

Q(z) = ε(z,z)

is always pure imaginary.

Generalization

A generalization called a semi-bilinear form was used by Reinhold Baer to characterize linear manifolds that are dual to each other in chapter 5 of his book Linear Algebra and Projective Geometry (1952). For a field F and A linear over F he requires

A pair consisting of an anti-automorphism α of the field F and a function f:A×A→F satisfying

for all a,b,c ∈ A $f(a+b,c)=f(a,c)+f(b,c),f(a,b+c)=f(a,b)+f(a,c),$ and

for all t ∈ F, all x,y ∈ A $f(tx,y)=tf(x,y),f(x,ty)=f(x,y)t^{\alpha }$ (page 101)

(The "transformation exponential notation" $t\mapsto t^{\alpha }$ is adopted in group theory literature.)

Baer calls such a form an α-form over A. The usual sesquilinear form has complex conjugation for α. When α is the identity, then f is a bilinear form.

In the algebraic structure called a *-ring the anti-automorphism is denoted by * and forms are constructed as indicated for α. Special constructions such as skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms are all considered in the broader context.

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

References

• K.W. Gruenberg & A.J. Weir (1977) Linear Geometry, §5.8 Sesquilinear Forms, pp 120–4, Springer, ISBN 0-387-90227-9 .
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