Sesquilinear form

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In mathematics, a sesquilinear form on a complex vector space V is a map V × VC 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[edit]

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[citation needed] and originates in Dirac's bra–ket notation in quantum mechanics. The opposite convention is more common in mathematics[citation needed].

Specifically a map φ : V × VC is sesquilinear if

&\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,wV and all a, bC. \bar a is the complex conjugate of a.

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[edit]

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[edit]

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 × VC 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

\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 {ei} 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 Hij = h(ei, ej).

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 zV.

Skew-Hermitian form[edit]

A skew-Hermitian form (also called an antisymmetric sesquilinear form), is a sesquilinear form ε : V × VC 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 {ei} 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 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×AF satisfying
for all a,b,cA f(a+b,c) = f(a,c) + f(b,c),\quad f(a,b+c) = f(a,b) + f(a,c), and
for all tF, all x,yA f(t x,y) = t f(x,y),\quad f(x,t y) = 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 \epsilon=\pm 1, to refer to both symmetric and skew-symmetric forms.