In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (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 B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on \R^n is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.

Coordinate representation

Let V be an n-dimensional vector space with basis {e1, …, en} . The n × n matrix A, defined by Aij = B(ei, ej) is called the matrix of the bilinear form on the basis {e1, …, en} . If the n × 1 matrix x represents a vector x with respect to this basis, and similarly, the n × 1 matrix y represents another vector y , 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 {f1, …, fn} is another basis of V, then where the S_{i,j} form an invertible matrix S. Then, the matrix of the bilinear form on the new basis is STAS .

Properties

Non-degenerate bilinear forms

Every bilinear form B on V defines a pair of linear maps from V to its dual space V∗ . Define B1, B2: V → V∗ 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). For a finite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear form B 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: x = 0 and y = 0 . The corresponding notion**** for a module**** over**** a commutative ring**** is**** that**** a bilinear**** form**** is**** **** if**** V → V∗ 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 B(x, y) = 2xy is nondegenerate but not unimodular, as the induced map from V = Z to V∗ = Z is multiplication by 2. 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 The left radical and right radical of the form B are the kernels of B1 and B2 respectively; they are the vectors orthogonal to the whole space on the left and on the right. If V is finite-dimensional then the rank of B1 is equal to the rank of B2 . If this number is equal to dim(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 left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the definition of nondegeneracy: 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 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 B(x, y) = 2xy over the integers.

Symmetric, skew-symmetric, and alternating forms

We define a bilinear form to be B(v, w) = B(w, v) for all v , w in V; B(v, v) = 0 for all v in V; B(v, w) = −B(w, v) for all v , w in V; B(v + w, v + w) . If the characteristic of K is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if char(K) = 2 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 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 char(K) ≠ 2 ). A bilinear form is symmetric if and only if the maps B1, B2: V → V∗ are equal, and skew-symmetric if and only if they are negatives of one another. If char(K) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows where tB is the transpose of B (defined above).

Reflexive bilinear forms and orthogonal vectors

A bilinear form B 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 v , with matrix representation x , is in the radical of a bilinear form with matrix representation A , if and only if Ax = 0 ⇔ xTA = 0 . 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 the orthogonal complement For a non-degenerate form on a finite-dimensional space, the map V/W → W⊥ is bijective, and the dimension of W⊥ is dim(V) − dim(W) .

Bounded and elliptic bilinear forms

Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is bounded, if there is a constant C such that for all u, v ∈ V , Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is elliptic, or coercive, if there is a constant c > 0 such that for all u ∈ V ,

Associated quadratic form

For any bilinear form B : V × V → K , there exists an associated quadratic form Q : V → K defined by Q : V → K : v ↦ B(v, v) . When char(K) ≠ 2 , 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 char(K) = 2 and dim V > 1 , this correspondence between quadratic forms and symmetric bilinear forms breaks down.

Relation to tensor products

By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on V and linear maps V ⊗ V → K . If B is a bilinear form on V the corresponding linear map is given by In the other direction, if F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w . The set of all linear maps V ⊗ V → K is the dual space of V ⊗ V , so bilinear forms may be thought of as elements of (V ⊗ V)∗ which (when V is finite-dimensional) is canonically isomorphic to V∗ ⊗ V∗ . Likewise, symmetric bilinear forms may be thought of as elements of (Sym2V)* (dual of the second symmetric power of V ) and alternating bilinear forms as elements of (Λ2V)∗ ≃ Λ2V∗ (the second exterior power of V∗ ). If charK ≠ 2 , (Sym2V)* ≃ Sym2(V∗) .

Generalizations

Pairs of distinct vector 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 V to W∗ , and from W to V∗ . 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 Z × Z → Z via (x, y) ↦ 2xy is nondegenerate, but induces multiplication by 2 on the map Z → Z∗ . 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 Aij 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 K, the instances with real numbers R , complex numbers C , and quaternions H are spelled out. The bilinear form is called the real symmetric case and labeled R(p, q) , where p + q = n . Then he articulates the connection to traditional terminology: "Some of the real symmetric cases are very important. The positive definite case R(n, 0) is called Euclidean space, while the case of a single minus, R(n−1, 1) is called Lorentzian space. If n = 4, then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case R(p, p) will be referred to as the split-case."

General modules

Given a ring R and a right R-module M and its dual module M∗ , a mapping B : M∗ × M → R is called a bilinear form if for all u, v ∈ M∗ , all x, y ∈ M and all α, β ∈ R . The mapping ⟨⋅,⋅⟩ : M∗ × M → R : (u, x) ↦ u(x) is known as the natural pairing, also called the canonical bilinear form on M∗ × M . A linear map S : M∗ → M∗ : u ↦ S(u) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨S(u), x⟩ , and a linear map T : M → M : x ↦ T(x) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨u, T(x)⟩ . Conversely, a bilinear form B : M∗ × M → R induces the R-linear maps S : M∗ → M∗ : u ↦ (x ↦ B(u, x)) and T′ : M → M∗∗ : x ↦ (u ↦ B(u, x)) . Here, M∗∗ denotes the double dual of M .

Citations