Sesquilinear form

Template:Short description In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.

A motivating special case is a sesquilinear form on a complex vector space, Template:Math. This is a map Template:Math that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism.

An application in projective geometry requires that the scalars come from a division ring (skew field), Template:Math, and this means that the "vectors" should be replaced by elements of a [[R-module|Template:Math-module]]. In a very general setting, sesquilinear forms can be defined over Template:Math-modules for arbitrary rings Template:Math.

Informal introduction

Sesquilinear forms abstract and generalize the basic notion of a Hermitian form on complex vector space. Hermitian forms are commonly seen in physics, as the inner product on a complex Hilbert space. In such cases, the standard Hermitian form on Template:Math is given by

<math>\langle w,z \rangle = \sum_{i=1}^n \overline{w}_i z_i.</math>

where <math>\overline{w}_i</math> denotes the complex conjugate of <math>w_i ~.</math> This product may be generalized to situations where one is not working with an orthonormal basis for Template:Math, or even any basis at all. By inserting an extra factor of <math>i</math> into the product, one obtains the skew-Hermitian form, defined more precisely, below. There is no particular reason to restrict the definition to the complex numbers; it can be defined for arbitrary rings carrying an antiautomorphism, informally understood to be a generalized concept of "complex conjugation" for the ring.

Convention

Conventions differ as to which argument should be linear. In the commutative case, we shall take the first to be linear, as is common in the mathematical literature, except in the section devoted to sesquilinear forms on complex vector spaces. There we use the other convention and take the first argument to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used mostly by physicists<ref>footnote 1 in Anthony Knapp Basic Algebra (2007) pg. 255</ref> and originates in Dirac's bra–ket notation in quantum mechanics. It is also consistent with the definition of the usual (Euclidean) product of <math>w,z\in\mathbb C^n</math> as <math>w^*z</math>.

In the more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear.

Complex vector spaces

Assumption: In this section, sesquilinear forms are antilinear in their first argument and linear in their second.

Over a complex vector space <math>V</math> a map <math>\varphi : V \times V \to \Complex</math> is sesquilinear if

<math>\begin{align}

&\varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w)\\ &\varphi(a x, b y) = \overline{a}b\,\varphi(x,y)\end{align}</math> for all <math>x, y, z, w \in V</math> and all <math>a, b \in \Complex.</math> Here, <math>\overline{a}</math> is the complex conjugate of a scalar <math>a.</math>

A complex sesquilinear form can also be viewed as a complex bilinear map <math display="block">\overline{V} \times V \to \Complex</math> where <math>\overline{V}</math> is the complex conjugate vector space to <math>V.</math> By the universal property of tensor products these are in one-to-one correspondence with complex linear maps <math display="block">\overline{V} \otimes V \to \Complex.</math>

For a fixed <math>z \in V</math> the map <math>w \mapsto \varphi(z, w)</math> is a linear functional on <math>V</math> (i.e. an element of the dual space <math>V^*</math>). Likewise, the map <math>w \mapsto \varphi(w, z)</math> is a conjugate-linear functional on <math>V.</math>

Given any complex sesquilinear form <math>\varphi</math> on <math>V</math> we can define a second complex sesquilinear form <math>\psi</math> via the conjugate transpose: <math display="block">\psi(w,z) = \overline{\varphi(z,w)}.</math> In general, <math>\psi</math> and <math>\varphi</math> will be different. If they are the same then <math>\varphi</math> is said to be Template:Em. If they are negatives of one another, then <math>\varphi</math> is said to be Template:Em. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.

Matrix representation

If <math>V</math> is a finite-dimensional complex vector space, then relative to any basis <math>\left\{ e_i \right\}_i</math> of <math>V,</math> a sesquilinear form is represented by a matrix <math>A,</math> and given by <math display="block">\varphi(w,z) = \varphi \left(\sum_i w_i e_i, \sum_j z_j e_j \right) = \sum_i \sum_j \overline{w_i} z_j \varphi\left(e_i, e_j\right) = w^\dagger A z .</math> where <math>w^\dagger</math> is the conjugate transpose. The components of the matrix <math>A</math> are given by <math>A_{ij} := \varphi\left(e_i, e_j\right).</math>

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 complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form <math>h : V \times V \to \Complex</math> such that <math display="block">h(w,z) = \overline{h(z, w)}.</math> The standard Hermitian form on <math>\Complex^n</math> is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by <math display="block">\langle w,z \rangle = \sum_{i=1}^n \overline{w}_i z_i.</math> More generally, the inner product on any complex Hilbert space is a Hermitian form.

A minus sign is introduced in the Hermitian form <math>w w^* - z z^*</math> to define the group SU(1,1).

A vector space with a Hermitian form <math>(V, h)</math> is called a Hermitian space.

The matrix representation of a complex Hermitian form is a Hermitian matrix.

A complex Hermitian form applied to a single vector <math display="block">|z|_h = h(z, z)</math> is always a real number. One can show that a complex sesquilinear form is Hermitian if and only if the associated quadratic form is real for all <math>z \in V.</math>

Skew-Hermitian form

A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form <math>s : V \times V \to \Complex</math> such that <math display="block">s(w,z) = -\overline{s(z, w)}.</math> Every complex skew-Hermitian form can be written as the imaginary unit <math>i := \sqrt{-1}</math> times a Hermitian form.

The matrix representation of a complex skew-Hermitian form is a skew-Hermitian matrix.

A complex skew-Hermitian form applied to a single vector <math display="block">|z|_s = s(z, z)</math> is always a purely imaginary number.

Over a division ring

This section applies unchanged when the division ring Template:Math is commutative. More specific terminology then also applies: the division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions.

Definition

A Template:Math-sesquilinear form over a right Template:Math-module Template:Math is a bi-additive map Template:Math with an associated anti-automorphism Template:Math of a division ring Template:Math such that, for all Template:Math in Template:Math and all Template:Math in Template:Math,

<math>\varphi(x \alpha, y \beta) = \sigma(\alpha) \, \varphi(x, y) \, \beta .</math>

The associated anti-automorphism Template:Math for any nonzero sesquilinear form Template:Math is uniquely determined by Template:Math.

Orthogonality

Given a sesquilinear form Template:Math over a module Template:Math and a subspace (submodule) Template:Math of Template:Math, the orthogonal complement of Template:Math with respect to Template:Math is

<math>W^{\perp}=\{\mathbf{v} \in M \mid \varphi (\mathbf{v}, \mathbf{w})=0,\ \forall \mathbf{w}\in W\} . </math>

Similarly, Template:Math is orthogonal to Template:Math with respect to Template:Math, written Template:Math (or simply Template:Math if Template:Math can be inferred from the context), when Template:Math. This relation need not be symmetric, i.e. Template:Math does not imply Template:Math (but see Template:Section link below).

Reflexivity

A sesquilinear form Template:Math is reflexive if, for all Template:Math in Template:Math,

<math>\varphi(x, y) = 0</math> implies <math>\varphi(y, x) = 0.</math>

That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric.

Hermitian variations

A Template:Math-sesquilinear form Template:Math is called Template:Math-Hermitian if there exists Template:Math in Template:Math such that, for all Template:Math in Template:Math,

<math>\varphi(x, y) = \sigma ( \varphi (y, x)) \, \varepsilon .</math>

If Template:Math, the form is called Template:Math-Hermitian, and if Template:Math, it is called Template:Math-anti-Hermitian. (When Template:Math is implied, respectively simply Hermitian or anti-Hermitian.)

For a nonzero Template:Math-Hermitian form, it follows that for all Template:Math in Template:Math,

<math> \sigma ( \varepsilon ) = \varepsilon^{-1} </math>

<math> \sigma ( \sigma ( \alpha ) ) = \varepsilon \alpha \varepsilon^{-1} .</math>

It also follows that Template:Math is a fixed point of the map Template:Math. The fixed points of this map form a subgroup of the additive group of Template:Math.

A Template:Math-Hermitian form is reflexive, and every reflexive Template:Math-sesquilinear form is Template:Math-Hermitian for some Template:Math.<ref> Template:Citation – [1] </ref><ref> Sesquilinear form at the Encyclopedia of Mathematics</ref><ref>Template:Citation – [2] </ref><ref name=Demb42> Template:Harvnb </ref>

In the special case that Template:Math is the identity map (i.e., Template:Math), Template:Math is commutative, Template:Math is a bilinear form and Template:Math. Then for Template:Math the bilinear form is called symmetric, and for Template:Math is called skew-symmetric.<ref>When Template:Math, skew-symmetric and symmetric bilinear forms coincide since then Template:Math. In all cases, alternating bilinear forms are a subset of skew-symmetric bilinear forms, and need not be considered separately.</ref>

Example

Let Template:Math be the three dimensional vector space over the finite field Template:Math, where Template:Math is a prime power. With respect to the standard basis we can write Template:Math and Template:Math and define the map Template:Math by:

<math>\varphi(x, y) = x_1 y_1{}^q + x_2 y_2{}^q + x_3 y_3{}^q.</math>

The map Template:Math is an involutory automorphism of Template:Math. The map Template:Math is then a Template:Math-sesquilinear form. The matrix Template:Math associated to this form is the identity matrix. This is a Hermitian form.

In projective geometry

Assumption: In this section, sesquilinear forms are antilinear (resp. linear) in their second (resp. first) argument.

In a projective geometry Template:Math, a permutation Template:Math of the subspaces that inverts inclusion, i.e.

Template:Math for all subspaces Template:Math, Template:Math of Template:Math,

is called a correlation. A result of Birkhoff and von Neumann (1936)<ref>Template:Citation</ref> shows that the correlations of desarguesian projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space.<ref name=Demb42 /> A sesquilinear form Template:Math is nondegenerate if Template:Math for all Template:Math in Template:Math (if and) only if Template:Math.

To achieve full generality of this statement, and since every desarguesian projective geometry may be coordinatized by a division ring, Reinhold Baer extended the definition of a sesquilinear form to a division ring, which requires replacing vector spaces by [[Module (mathematics)|Template:Math-module]]s.<ref>Template:Citation</ref> (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.)<ref>Baer's terminology gives a third way to refer to these ideas, so he must be read with caution.</ref>

Over arbitrary rings

The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to the nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the arbitrary field version of the definition to arbitrary rings.

Let Template:Math be a ring, Template:Math an Template:Math-module and Template:Math an antiautomorphism of Template:Math.

A map Template:Math is Template:Math-sesquilinear if

<math>\varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w)</math>

<math>\varphi(c x, d y) = c \, \varphi(x,y) \, \sigma(d)</math>

for all Template:Math in Template:Math and all Template:Math in Template:Math.

An element Template:Math is orthogonal to another element Template:Math with respect to the sesquilinear form Template:Math (written Template:Math) if Template:Math. This relation need not be symmetric, i.e. Template:Math does not imply Template:Math.

A sesquilinear form Template:Math is reflexive (or orthosymmetric) if Template:Math implies Template:Math for all Template:Math in Template:Math.

A sesquilinear form Template:Math is Hermitian if there exists Template:Math such that<ref>Template:Citation</ref>Template:Rp

<math>\varphi(x, y) = \sigma(\varphi(y, x))</math>

for all Template:Math in Template:Math. A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism Template:Math is an involution (i.e. of order 2).

Since for an antiautomorphism Template:Math we have Template:Math for all Template:Math in Template:Math, if Template:Math, then Template:Math must be commutative and Template:Math is a bilinear form. In particular, if, in this case, Template:Math is a skewfield, then Template:Math is a field and Template:Math is a vector space with a bilinear form.

An antiautomorphism Template:Math can also be viewed as an isomorphism Template:Math, where Template:Math is the opposite ring of Template:Math, which has the same underlying set and the same addition, but whose multiplication operation (Template:Math) is defined by Template:Math, where the product on the right is the product in Template:Math. It follows from this that a right (left) Template:Math-module Template:Math can be turned into a left (right) Template:Math-module, Template:Math.<ref>Template:Harvnb</ref> Thus, the sesquilinear form Template:Math can be viewed as a bilinear form Template:Math.

See also

• *-ring

References

• Template:Citation
• Template:Citation
• Template:Citation

External links

• Template:Springer