In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a bilinear form is a
bilinear map on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(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
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 argument separately:
* and
* and
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 ...
on
is an example of a bilinear form.
The definition of a bilinear form can be extended to include
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
ring, with
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s replaced by
module homomorphisms.
When is the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s , one is often more interested in
sesquilinear forms, which are similar to bilinear forms but are
conjugate linear in one argument.
Coordinate representation
Let be an -
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space with
basis .
The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis .
If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then:
A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all
congruent. More precisely, if is another basis of , then
where the
form an
invertible matrix . Then, the matrix of the bilinear form on the new basis is .
Maps to the dual space
Every bilinear form on defines a pair of linear maps from to its
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 cons ...
. Define by
This is often denoted as
where the dot ( ⋅ ) indicates the slot into which the argument for the resulting
linear functional is to be placed (see
Currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f tha ...
).
For a finite-dimensional vector space , if either of or is an isomorphism, then both are, and the bilinear form is said to be
nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
:
for all
implies that and
:
for all
implies that .
The corresponding notion for a module over a commutative ring is that a bilinear form is if is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing is nondegenerate but not unimodular, as the induced map from to is multiplication by 2.
If is finite-dimensional then one can identify with its double dual . One can then show that is the
transpose of the linear map (if is infinite-dimensional then is the transpose of restricted to the image of in ). Given one can define the ''transpose'' of to be the bilinear form given by
The left radical and right radical of the form are the
kernels of and respectively; they are the vectors orthogonal to the whole space on the left and on the right.
If is finite-dimensional then the
rank of is equal to the rank of . If this number is equal to then and are linear isomorphisms from to . In this case is nondegenerate. By the
rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the ''definition'' of nondegeneracy:
Given any linear map one can obtain a bilinear form ''B'' on ''V'' via
This form will be nondegenerate if and only if is an isomorphism.
If is
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
then, relative to some
basis for , a bilinear form is degenerate if and only if the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is
non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a
unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example over the integers.
Symmetric, skew-symmetric and alternating forms
We define a bilinear form to be
*
symmetric if for all , in ;
*
alternating if for all in ;
* or if for all , in ;
*; Proposition: Every alternating form is skew-symmetric.
*; Proof: This can be seen by expanding .
If the
characteristic of is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.
A bilinear form is symmetric (respectively skew-symmetric)
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 bic ...
its coordinate matrix (relative to any basis) is
symmetric (respectively
skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when ).
A bilinear form is symmetric if and only if the maps are equal, and skew-symmetric if and only if they are negatives of one another. If then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
where is the transpose of (defined above).
Derived quadratic form
For any bilinear form , there exists an 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 ...
defined by .
When , the quadratic form ''Q'' is determined by the symmetric part of the bilinear form ''B'' and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.
When and , this correspondence between quadratic forms and symmetric bilinear forms breaks down.
Reflexivity and orthogonality
A bilinear form is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the ''kernel'' or the ''radical'' of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector , with matrix representation , is in the radical of a bilinear form with matrix representation , if and only if . The radical is always a subspace of . It is trivial if and only if the matrix is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Suppose is a subspace. Define the ''
orthogonal complement''
For a non-degenerate form on a finite-dimensional space, the map is
bijective, and the dimension of is .
Different spaces
Much of the theory is available for a
bilinear mapping from two vector spaces over the same base field to that field
Here we still have induced linear mappings from to , and from to . It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, ''B'' is said to be a perfect pairing.
In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance via is nondegenerate, but induces multiplication by 2 on the map .
Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product". To define them he uses diagonal matrices ''A
ij'' having only +1 or −1 for non-zero elements. Some of the "inner products" are
symplectic forms and some are
sesquilinear forms or
Hermitian forms. Rather than a general field , the instances with real numbers , complex numbers , and
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
are spelled out. The bilinear form
is called the real symmetric case and labeled , where . Then he articulates the connection to traditional terminology:
Relation to tensor products
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 fr ...
of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same 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 element of V \otime ...
, there is a canonical correspondence between bilinear forms on and linear maps . If is a bilinear form on the corresponding linear map is given by
In the other direction, if is a linear map the corresponding bilinear form is given by composing ''F'' with the bilinear map that sends to .
The set of all linear maps is 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 cons ...
of , so bilinear forms may be thought of as elements of which (when is finite-dimensional) is canonically isomorphic to .
Likewise, symmetric bilinear forms may be thought of as elements of (the second
symmetric power of ), and alternating bilinear forms as elements of (the second
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of ).
On normed vector spaces
Definition: A bilinear form on a
normed vector space is bounded, if there is a constant such that for all ,
Definition: A bilinear form on a normed vector space is elliptic, or
coercive, if there is a constant such that for all ,
Generalization to modules
Given a
ring and a right
-module and its
dual module , a mapping is called a bilinear form if
for all , all and all .
The mapping is known as the ''
natural pairing
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb.
Duality theory, the study of dua ...
'', also called the ''canonical bilinear form'' on .
A linear map induces the bilinear form , and a linear map induces the bilinear form .
Conversely, a bilinear form induces the ''R''-linear maps and . Here, denotes the
double dual
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 co ...
of .
See also
Citations
References
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* . Also:
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External links
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{{PlanetMath attribution, id=7553, title=Unimodular
Abstract algebra
Linear algebra
Multilinear algebra