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.

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

[edit] 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 is sesquilinear if

for all x,y,z,w ∈ V and all a, b ∈ C.

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

where 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

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

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.

[edit] 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) = i²B(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 .

[edit] 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

The standard Hermitian form on Cⁿ is given by

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:

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.

[edit] 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.

[edit] 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 , to refer to both symmetric and skew-symmetric forms.

[edit] References

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