Inmathematics, abilinear form is abilinear mapV ×V →K on avector spaceV (the elements of which are calledvectors) over afieldK (the elements of which are calledscalars). In other words, a bilinear form is a functionB :V ×V →K that islinear in each argument separately:
Thedot product on is an example of a bilinear form which is also aninner product.[1] An example of a bilinear form that is not an inner product would be thefour-vector product.
The definition of a bilinear form can be extended to includemodules over aring, withlinear maps replaced bymodule homomorphisms.
WhenK is the field ofcomplex numbersC, one is often more interested insesquilinear forms, which are similar to bilinear forms but areconjugate linear in one argument.
LetV be ann-dimensional vector space withbasis{e1, …,en}.
Then × n matrixA, defined byAij =B(ei,ej) is called thematrix of the bilinear form on the basis{e1, …,en}.
If then × 1 matrixx represents a vectorx with respect to this basis, and similarly, then × 1 matrixy represents another vectory, then:
A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are allcongruent. More precisely, if{f1, …,fn} is another basis ofV, thenwhere the form aninvertible matrixS. Then, the matrix of the bilinear form on the new basis isSTAS.
Dot product is represented by then × nidentity matrix.
Every bilinear formB onV defines a pair of linear maps fromV to itsdual spaceV∗. DefineB1,B2:V →V∗ by
This is often denoted as
where the dot ( ⋅ ) indicates the slot into which the argument for the resultinglinear functional is to be placed (seeCurrying).
For a finite-dimensional vector spaceV, if either ofB1 orB2 is an isomorphism, then both are, and the bilinear formB is said to benondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
The corresponding notion for a module over a commutative ring is that a bilinear form isunimodular ifV →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 pairingB(x,y) = 2xy is nondegenerate but not unimodular, as the induced map fromV =Z toV∗ =Z is multiplication by 2.
IfV is finite-dimensional then one can identifyV with its double dualV∗∗. One can then show thatB2 is thetranspose of the linear mapB1 (ifV is infinite-dimensional thenB2 is the transpose ofB1 restricted to the image ofV inV∗∗). GivenB one can define thetranspose ofB to be the bilinear form given by
Theleft radical andright radical of the formB are thekernels ofB1 andB2 respectively;[2] they are the vectors orthogonal to the whole space on the left and on the right.[3]
IfV is finite-dimensional then therank ofB1 is equal to the rank ofB2. If this number is equal todim(V) thenB1 andB2 are linear isomorphisms fromV toV∗. In this caseB is nondegenerate. By therank–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 thedefinition of nondegeneracy:
Given any linear mapA :V →V∗ one can obtain a bilinear formB onV via
This form will be nondegenerate if and only ifA is an isomorphism.
IfV isfinite-dimensional then, relative to somebasis forV, a bilinear form is degenerate if and only if thedeterminant 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 isnon-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 aunit (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 exampleB(x,y) = 2xy over the integers.
We define a bilinear form to be
If thecharacteristic ofK is not 2 then the converse is also true: every skew-symmetric form is alternating. However, ifchar(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) issymmetric (respectivelyskew-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 whenchar(K) ≠ 2).
A bilinear form is symmetric if and only if the mapsB1,B2:V →V∗ are equal, and skew-symmetric if and only if they are negatives of one another. Ifchar(K) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as followswheretB is the transpose ofB (defined above).
A bilinear formB is reflexive if and only if it is either symmetric or alternating.[4] 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 thekernel or theradical of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vectorv, with matrix representationx, is in the radical of a bilinear form with matrix representationA, if and only ifAx = 0 ⇔xTA = 0. The radical is always a subspace ofV. It is trivial if and only if the matrixA is nonsingular, and thus if and only if the bilinear form is nondegenerate.
SupposeW is a subspace. Define theorthogonal complement[5]
For a non-degenerate form on a finite-dimensional space, the mapV/W →W⊥ isbijective, and the dimension ofW⊥ isdim(V) − dim(W).
Definition: A bilinear form on anormed vector space(V, ‖⋅‖) isbounded, if there is a constantC such that for allu,v ∈V,
Definition: A bilinear form on a normed vector space(V, ‖⋅‖) iselliptic, orcoercive, if there is a constantc > 0 such that for allu ∈V,
For any bilinear formB :V ×V →K, there exists an associatedquadratic formQ :V →K defined byQ :V →K :v ↦B(v,v).
Whenchar(K) ≠ 2, the quadratic formQ is determined by the symmetric part of the bilinear formB 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.
Whenchar(K) = 2 anddimV > 1, this correspondence between quadratic forms and symmetric bilinear forms breaks down.
By theuniversal property of thetensor product, there is a canonical correspondence between bilinear forms onV and linear mapsV ⊗V →K. IfB is a bilinear form onV the corresponding linear map is given by
In the other direction, ifF :V ⊗V →K is a linear map the corresponding bilinear form is given by composingF with the bilinear mapV ×V →V ⊗V that sends(v,w) tov⊗w.
The set of all linear mapsV ⊗V →K is thedual space ofV ⊗V, so bilinear forms may be thought of as elements of(V ⊗V)∗ which (whenV is finite-dimensional) is canonically isomorphic toV∗ ⊗V∗.
Likewise, symmetric bilinear forms may be thought of as elements of(Sym2V)* (dual of the secondsymmetric power ofV) and alternating bilinear forms as elements of(Λ2V)∗ ≃ Λ2V∗ (the secondexterior power ofV∗). Ifchar(K) ≠ 2,(Sym2V)* ≃ Sym2(V∗).
Much of the theory is available for abilinear mapping from two vector spaces over the same base field to that field
Here we still have induced linear mappings fromV toW∗, and fromW toV∗. 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 aperfect 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 instanceZ ×Z →Z via(x,y) ↦ 2xy is nondegenerate, but induces multiplication by 2 on the mapZ →Z∗.
Terminology varies in coverage of bilinear forms. For example,F. Reese Harvey discusses "eight types of inner product".[6] To define them he uses diagonal matricesAij having only +1 or −1 for non-zero elements. Some of the "inner products" aresymplectic forms and some aresesquilinear forms orHermitian forms. Rather than a general fieldK, the instances with real numbersR, complex numbersC, andquaternionsH are spelled out. The bilinear formis called thereal symmetric case and labeledR(p,q), wherep +q =n. Then he articulates the connection to traditional terminology:[7]
Some of the real symmetric cases are very important. The positive definite caseR(n, 0) is calledEuclidean space, while the case of a single minus,R(n−1, 1) is calledLorentzian space. Ifn = 4, then Lorentzian space is also calledMinkowski space orMinkowski spacetime. The special caseR(p,p) will be referred to as thesplit-case.
Given aringR and a rightR-moduleM and itsdual moduleM∗, a mappingB :M∗ ×M →R is called abilinear form if
for allu,v ∈M∗, allx,y ∈M and allα,β ∈R.
The mapping⟨⋅,⋅⟩ :M∗ ×M →R : (u,x) ↦u(x) is known as thenatural pairing, also called thecanonical bilinear form onM∗ ×M.[8]
A linear mapS :M∗ →M∗ :u ↦S(u) induces the bilinear formB :M∗ ×M →R : (u,x) ↦ ⟨S(u),x⟩, and a linear mapT :M →M :x ↦T(x) induces the bilinear formB :M∗ ×M →R : (u,x) ↦ ⟨u,T(x)⟩.
Conversely, a bilinear formB :M∗ ×M →R induces theR-linear mapsS :M∗ →M∗ :u ↦ (x ↦B(u,x)) andT′ :M →M∗∗ :x ↦ (u ↦B(u,x)). Here,M∗∗ denotes thedouble dual ofM.
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This article incorporates material from Unimodular onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
