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Generalization of a bilinear form
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 prefixsesqui- 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, V. This is a map V × V -> C 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), K, and this means that the "vectors" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary ringsR.
where denotes the complex conjugate of This product may be generalized to situations where one is not working with an orthonormal basis for Cn, or even any basis at all. By inserting an extra factor of 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.
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 and originates in Dirac'sbra-ket notation in quantum mechanics.
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.
Given any complex sesquilinear form on we can define a second complex sesquilinear form via the conjugate transpose:
In general, and will be different. If they are the same then is said to be Hermitian. If they are negatives of one another, then is said to be skew-Hermitian. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.
If is a finite-dimensional complex vector space, then relative to any basis of a sesquilinear form is represented by a matrix by the column vector and by the column vector :
This section applies unchanged when the division ring K 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.
The associated anti-automorphism ? for any nonzero sesquilinear form ? is uniquely determined by ?.
Given a sesquilinear form ? over a module M and a subspace (submodule) W of M, the orthogonal complement of W with respect to ? is
Similarly, x ? M is orthogonal to y ? M with respect to ?, written x ??y (or simply x ? y if ? can be inferred from the context), when ?(x, y) = 0. This relation need not be symmetric, i.e. x ? y does not imply y ? x (but see § Reflexivity below).
A sesquilinear form ? is reflexive if, for all x, y in M,
That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric.
A ?-sesquilinear form ? is called (?, ?)-Hermitian if there exists ? in K such that, for all x, y in M,
If ? = 1, the form is called ?-Hermitian, and if ? = -1, it is called ?-anti-Hermitian. (When ? is implied, respectively simply Hermitian or anti-Hermitian.)
For a nonzero (?, ?)-Hermitian form, it follows that for all ? in K,
A (?, ?)-Hermitian form is reflexive, and every reflexive ?-sesquilinear form is (?, ?)-Hermitian for some ?.
In the special case that ? is the identity map (i.e., ? = id), K is commutative, ? is a bilinear form and ?2 = 1. Then for ? = 1 the bilinear form is called symmetric, and for ? = -1 is called skew-symmetric.
Let V be the three dimensional vector space over the finite fieldF = GF(q2), where q is a prime power. With respect to the standard basis we can write x = (x1, x2, x3) and y = (y1, y2, y3) and define the map ? by:
The map ? : t ? tq is an involutory automorphism of F. The map ? is then a ?-sesquilinear form. The matrix M? 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.
is called a correlation. A result of Birkhoff and von Neumann (1936) shows that the correlations of desarguesian projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space. A sesquilinear form ? is nondegenerate if ?(x, y) = 0 for all y in V (if and) only if x = 0.
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 R-modules. (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.)
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.
An element x is orthogonal to another element y with respect to the sesquilinear form ? (written x ? y) if ?(x, y) = 0. This relation need not be symmetric, i.e. x ? y does not imply y ? x.
A sesquilinear form ? : V × V -> R is reflexive (or orthosymmetric) if ?(x, y) = 0 implies ?(y, x) = 0 for all x, y in V.
A sesquilinear form ? : V × V -> R is Hermitian if there exists ? such that
for all x, y in V. A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism ? is an involution (i.e. of order 2).
Since for an antiautomorphism ? we have ?(st) = ?(t)?(s) for all s, t in R, if ? = id, then R must be commutative and ? is a bilinear form. In particular, if, in this case, R is a skewfield, then R is a field and V is a vector space with a bilinear form.
An antiautomorphism ? : R -> R can also be viewed as an isomorphismR -> Rop, where Rop is the opposite ring of R, which has the same underlying set and the same addition, but whose multiplication operation (*) is defined by a * b = ba, where the product on the right is the product in R. It follows from this that a right (left) R-module V can be turned into a left (right) Rop-module, Vo. Thus, the sesquilinear form ? : V × V -> R can be viewed as a bilinear form ?? : V × Vo -> R.
^When charK = 2, skew-symmetric and symmetric bilinear forms coincide since then 1 = -1. In all cases, alternating bilinear forms are a subset of skew-symmetric bilinear forms, and need not be considered separately.