Bilinear form
Encyclopedia
In mathematics
, a bilinear form on a vector space
V is a bilinear mapping V × V → F, where F is the field
of scalar
s. That is, a bilinear form is a function B: V × V → F which is linear
in each argument separately:
Any bilinear form on
can be expressed as
where A is an n × n matrix.
The definition of a bilinear form can easily be extended to include module
s over a commutative ring
, with linear maps replaced by module homomorphisms. When F is the field of complex number
s C, one is often more interested in sesquilinear form
s, which are similar to bilinear forms but are conjugate linear in one argument.
be a basis for a finite-dimensional space V. Define the 
- matrix A by 
. Then if the 
matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:
Suppose C' is another basis for V, with :
with S an invertible
- matrix.
Now the new matrix representation for the symmetric bilinear form is given by :
V*. Define
by
This is often denoted as
where the (
) indicates the slot into which the argument for the resulting linear functional
is to be placed.
If either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate
. This can only occur if V is finite-dimensional since V* has higher dimension than V otherwise.
If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B2 is the transpose
of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V**). Given B one can define the transpose of B to be the bilinear form given by
If V is finite-dimensional then the rank
of B1 is equal to the rank of B2. If this number is equal to the dimension of V then B1 and B2 are linear isomorphisms from V to V*. In this case B is nondegenerate. By the rank-nullity theorem
, this is equivalent to the condition that the kernel
of B1 be trivial. In fact, for finite dimensional spaces, this is often taken as the definition of nondegeneracy. Thus B is nondegenerate if and only if
Given any linear map A : V → V* one can obtain a bilinear form B on V via
This form will be nondegenerate if and only if A is an isomorphism.
If V is finite-dimensional then, relative to some basis
for V, a bilinear form is degenerate if and only if the determinant
of the associated matrix is zero. Likewise, a nondegenerate form is one for which the associated matrix is non-singular. These statements are independent of the chosen basis.
is reflexive if
Reflexivity allows us to define orthogonality: two vectors v and w are orthogonal with respect to the reflexive bilinear form if and only if :
or 
The radical of a bilinear form is the subset of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if :
The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Suppose W is a subspace. Define :
When the bilinear form is nondegenerate, the map
is bijective, and the dimension of 
is dim(V)-dim(W).
One can prove that B is reflexive if and only if it is either:
Every alternating form is skew-symmetric (
). This may be seen by expanding B(v+w,v+w).
If the characteristic
of F is not 2 then the converse is also true (every skew-symmetric form is alternating). If, however, char(F) = 2 then a skew-symmetric form is the same thing as a symmetric form and not all of these are alternating.
A bilinear form is symmetric (resp. skew-symmetric) if and only if
its coordinate matrix (relative to any basis) is symmetric (resp. 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 char(F) ≠ 2).
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 char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and a skew-symmetric part as follows
where B* is the transpose of B (defined above).
Also if char(F) ≠ 2 then one can define a quadratic form
in terms of its associated symmetric form. One can likewise define quadratic forms corresponding to skew-symmetric forms, Hermitian forms, and skew-Hermitian forms; the general concept is ε-quadratic form.
In this situation we still have linear mappings of V to the dual space of W, and of W to the dual space of V. It may happen that both of those 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), nondegenerate is a weaker notion: a pairing can be nondegenerate without being a perfect pairing, for instance
via 
is nondegenerate, but induces multiplication by 2 on the map 
of the tensor product
, bilinear forms on V are in 1-to-1 correspondence with linear maps V ⊗ V → F. If B is a bilinear form on V the corresponding linear map is given by
The set of all linear maps V ⊗ V → F is the dual space
of V ⊗ V, so bilinear forms may be thought of as elements of
Likewise, symmetric bilinear forms may be thought of as elements of S2V* (the second symmetric power of V*), and alternating bilinear forms as elements of Λ2V* (the second exterior power of V*).
is bounded, if there is a constant
such that for all 
A bilinear form on a normed vector space is elliptic, or coercive, if there is a constant
such that for all 
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a bilinear form on a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V is a bilinear mapping V × V → F, where F is the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
s. That is, a bilinear form is a function B: V × V → F which is linear
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
in each argument separately:
Any bilinear form on
can be expressed aswhere A is an n × n matrix.
The definition of a bilinear form can easily be extended to include module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s over a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, with linear maps replaced by module homomorphisms. When F is the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C, one is often more interested in sesquilinear form
Sesquilinear form
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...
s, which are similar to bilinear forms but are conjugate linear in one argument.
Coordinate representation
Let be a basis for a finite-dimensional space V. Define the - matrix A by . Then if the matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:Suppose C' is another basis for V, with :
with S an invertible
- matrix.Now the new matrix representation for the symmetric bilinear form is given by :
Maps to the dual space
Every bilinear form B on V defines a pair of linear maps from V to its dual spaceDual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
V*. Define
byThis is often denoted as
where the (
) indicates the slot into which the argument for the resulting linear functionalLinear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
is to be placed.
If either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate
Degenerate form
In mathematics, specifically linear algebra, a degenerate bilinear form ƒ on a vector space V is one such that the map from V to V^* given by v \mapsto is not an isomorphism...
. This can only occur if V is finite-dimensional since V* has higher dimension than V otherwise.
If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B2 is the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V**). Given B one can define the transpose of B to be the bilinear form given by
If V is finite-dimensional then the rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
of B1 is equal to the rank of B2. If this number is equal to the dimension of V then B1 and B2 are linear isomorphisms from V to V*. In this case B is nondegenerate. By the rank-nullity theorem
Rank-nullity theorem
In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix over some field, thenThis applies to linear maps as well...
, this is equivalent to the condition that the kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of B1 be trivial. In fact, for finite dimensional spaces, this is often taken as the definition of nondegeneracy. Thus B is nondegenerate if and only if
Given any linear map A : V → V* one can obtain a bilinear form B on V via
This form will be nondegenerate if and only if A is an isomorphism.
If V is finite-dimensional then, relative to some basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
for V, a bilinear form is degenerate if and only if the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the associated matrix is zero. Likewise, a nondegenerate form is one for which the associated matrix is non-singular. These statements are independent of the chosen basis.
Reflexivity and orthogonality
A bilinear form- B : V × V → F
is reflexive if
Reflexivity allows us to define orthogonality: two vectors v and w are orthogonal with respect to the reflexive bilinear form if and only if :
or The radical of a bilinear form is the subset of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if :
The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Suppose W is a subspace. Define :
When the bilinear form is nondegenerate, the map
is bijective, and the dimension of is dim(V)-dim(W).One can prove that B is reflexive if and only if it is either:
- symmetricSymmetric bilinear formA symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
:
for all 
; or - alternating if
for all 
Every alternating form is skew-symmetric (
). This may be seen by expanding B(v+w,v+w).If the characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
of F is not 2 then the converse is also true (every skew-symmetric form is alternating). If, however, char(F) = 2 then a skew-symmetric form is the same thing as a symmetric form and not all of these are alternating.
A bilinear form is symmetric (resp. skew-symmetric) if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its coordinate matrix (relative to any basis) is symmetric (resp. skew-symmetric
Skew-symmetric matrix
In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
). 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 char(F) ≠ 2).
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 char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and a skew-symmetric part as followswhere B* is the transpose of B (defined above).
Also if char(F) ≠ 2 then one can define a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
in terms of its associated symmetric form. One can likewise define quadratic forms corresponding to skew-symmetric forms, Hermitian forms, and skew-Hermitian forms; the general concept is ε-quadratic form.
Different spaces
Much of the theory is available for a bilinear mapping- B: V × W → F.
In this situation we still have linear mappings of V to the dual space of W, and of W to the dual space of V. It may happen that both of those 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), nondegenerate is a weaker notion: a pairing can be nondegenerate without being a perfect pairing, for instance
via is nondegenerate, but induces multiplication by 2 on the map Relation to tensor products
By the universal propertyUniversal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
of the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
, bilinear forms on V are in 1-to-1 correspondence with linear maps V ⊗ V → F. If B is a bilinear form on V the corresponding linear map is given by
The set of all linear maps V ⊗ V → F is the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of V ⊗ V, so bilinear forms may be thought of as elements of
Likewise, symmetric bilinear forms may be thought of as elements of S2V* (the second symmetric power of V*), and alternating bilinear forms as elements of Λ2V* (the second exterior power of V*).
On normed vector spaces
A bilinear form on a normed vector spaceNormed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
is bounded, if there is a constant
such that for all A bilinear form on a normed vector space is elliptic, or coercive, if there is a constant
such that for all See also
- Bilinear operatorBilinear operatorIn mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments. Matrix multiplication is an example.-Definition:...
- Multilinear form
- Quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
- Inner product spaceInner product spaceIn mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
- Positive semi definite
- Sesquilinear formSesquilinear formIn mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...