In
mathematics, a sesquilinear form is a generalization of a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is lin ...
that, in turn, is a generalization of the concept of the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. A bilinear form is
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the 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
Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example:
* unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cyc ...
''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, . This is a map 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
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
requires that the scalars come from a
division ring (skew field), , and this means that the "vectors" should be replaced by elements of a
-module. In a very general setting, sesquilinear forms can be defined over -modules for arbitrary
rings .
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, as the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on a complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
. In such cases, the standard Hermitian form on is given by
:
where
denotes the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of
This product may be generalized to situations where one is not working with an orthonormal basis for , 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.
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 and originates in
Dirac's bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathem ...
in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
.
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
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in their second.
Over a
complex vector space a map
is sesquilinear if
:
for all
and all
Here,
is the complex conjugate of a scalar
A complex sesquilinear form can also be viewed as a complex
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, ...
where
is the
complex conjugate vector space
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In oth ...
to
By the
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
of
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
s these are in one-to-one correspondence with complex linear maps
For a fixed
the map
is a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , th ...
on
(i.e. an element of the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
). Likewise, the map
is a
conjugate-linear functional on
Given any complex sesquilinear form
on
we can define a second complex sesquilinear form
via the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
:
In general,
and
will be different. If they are the same then
is said to be . If they are negatives of one another, then
is said to be . Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.
Matrix representation
If
is a finite-dimensional complex vector space, then relative to any
basis of
a sesquilinear form is represented by a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
and given by
where
is the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
. The components of the matrix
are given by
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
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
on a
Hermitian manifold.''
A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form
such that
The standard Hermitian form on
is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by
More generally, the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on any complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
is a Hermitian form.
A minus sign is introduced in the Hermitian form
to define the group
SU(1,1).
A vector space with a Hermitian form
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
is always a
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
. One can show that a complex sesquilinear form is Hermitian
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
the associated
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
is real for all
Skew-Hermitian form
A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form
such that
Every complex skew-Hermitian form can be written as the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
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
is always a purely
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
.
Over a division ring
This section applies unchanged when the division ring is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. 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 -sesquilinear form over a right -module is a
bi-additive map with an associated
anti-automorphism of a
division ring such that, for all in and all in ,
:
The associated anti-automorphism for any nonzero sesquilinear form is uniquely determined by .
Orthogonality
Given a sesquilinear form over a module and a subspace (
submodule) of , the orthogonal complement of with respect to is
:
Similarly, is orthogonal to with respect to , written (or simply if can be inferred from the context), when . This
relation need not be
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, i.e. does not imply (but see ' below).
Reflexivity
A sesquilinear form is reflexive if, for all in ,
:
implies
That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric.
Hermitian variations
A -sesquilinear form is called -Hermitian if there exists in such that, for all in ,
:
If , the form is called -''Hermitian'', and if , 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 ,
:
:
It also follows that is a
fixed point of the map . The fixed points of this map form a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structure ...
of .
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., ), is commutative, is a bilinear form and . Then for the bilinear form is called ''symmetric'', and for is called ''skew-symmetric''.
Example
Let be the three dimensional vector space over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
, where is a
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, ...
. With respect to the standard basis we can write and and define the map by:
:
The map is an
involutory automorphism of . The map is then a -sesquilinear form. The matrix associated to this form is the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
. This is a Hermitian form.
In projective geometry
:Assumption: In this section, sesquilinear forms are
antilinear (resp.
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
) in their second (resp. first) argument.
In a
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
, a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
of the subspaces that inverts inclusion, i.e.
: for all subspaces , of ,
is called a
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statisti ...
. 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 for all in (if and) only if .
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
-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.
Let be a
ring, an -
module and an
antiautomorphism of .
A map is -sesquilinear if
:
:
for all in and all in .
An element is orthogonal to another element with respect to the sesquilinear form (written ) if . This relation need not be symmetric, i.e. does not imply .
A sesquilinear form is reflexive (or ''orthosymmetric'') if implies for all in .
A sesquilinear form is Hermitian if there exists such that
:
for all in . 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 for all in , if , then must be commutative and is a bilinear form. In particular, if, in this case, is a skewfield, then is a field and is a vector space with a bilinear form.
An antiautomorphism can also be viewed as an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, where is the
opposite ring of , which has the same underlying set and the same addition, but whose multiplication operation () is defined by , where the product on the right is the product in . It follows from this that a right (left) -module can be turned into a left (right) -module, .
Thus, the sesquilinear form can be viewed as a bilinear form .
See also
*
*-ring
Notes
References
*
*
*
External links
*
{{Hilbert space
Functional analysis
Linear algebra