Bilinear form

Template:Short description In mathematics, a bilinear form is a bilinear map V × V → K on a vector space Template:Mvar (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 ${\mathbb{ℝ}}^n$ is an example of a bilinear form which is also an inner product. An example of a bilinear form that is not an inner product would be the four-vector product.

The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.

When Template:Mvar 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 Template:Mvar-dimensional vector space with basis {e₁, …, e_n}.

The n × n matrix A, defined by A_ij = B(e_i, e_j) is called the matrix of the bilinear form on the basis {e₁, …, e_n}.

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: $$B(\mathbf{𝐱},\mathbf{𝐲})={\mathbf{𝐱}}^{\text{T}}A\mathbf{𝐲}=\sum _{i,j=1}^n{x}_i{A}_{ij}{y}_j.$$

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, …, f_n} is another basis of Template:Mvar, then $${\mathbf{𝐟}}_j=\sum _{i=1}^n{S}_{i,j}{\mathbf{𝐞}}_i,$$ where the $S_{i,j}$ form an invertible matrix Template:Mvar. Then, the matrix of the bilinear form on the new basis is S^TAS.

Dot product is represented by the n × n identity matrix.

Properties

Non-degenerate bilinear forms

Every bilinear form B on Template:Mvar defines a pair of linear maps from Template:Mvar to its dual space V^∗. Define B₁, B₂: V → V^∗ by Template:Block indent Template:Block indent This is often denoted as Template:Block indent Template:Block indent 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 Template:Mvar, 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:

$B(x,y)=0$ for all $y\in V$ implies that x = 0 and

$B(x,y)=0$ for all $x\in V$ implies that y = 0.

The corresponding notion for a module over a commutative ring is that a bilinear form is Template:Visible anchor 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 Template:Mvar is finite-dimensional then one can identify Template:Mvar with its double dual V^∗∗. One can then show that B₂ is the transpose of the linear map B₁ (if Template:Mvar is infinite-dimensional then B₂ is the transpose of B₁ restricted to the image of Template:Mvar in V^∗∗). Given B one can define the transpose of B to be the bilinear form given by Template:Block indent

The left radical and right radical of the form B are the kernels of B₁ and B₂ respectively;Template:Sfn they are the vectors orthogonal to the whole space on the left and on the right.Template:Sfn

If Template:Mvar 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 Template:Mvar 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: Template:Block indent

Given any linear map A : V → V^∗ one can obtain a bilinear form B on V via Template:Block indent

This form will be nondegenerate if and only if A is an isomorphism.

If Template:Mvar is finite-dimensional then, relative to some basis for Template:Mvar, 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

• symmetric if B(v, w) = B(w, v) for all v, w in Template:Mvar;
• alternating if B(v, v) = 0 for all v in Template:Mvar;
• Template:Visible anchor or Template:Visible anchor if B(v, w) = −B(w, v) for all v, w in Template:Mvar;

  Proposition

  Every alternating form is skew-symmetric.

  Proof

  This can be seen by expanding B(v + w, v + w).

If the characteristic of Template:Mvar 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 $$B^{+}=\frac{1}{2}(B+{}^{\text{t}}B)B^{-}=\frac{1}{2}(B-{}^{\text{t}}B),$$ 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.Template:Sfn 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 ⇔ x^TA = 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 Template:Mvar is a subspace. Define the orthogonal complementTemplate:Sfn $$W^{\perp }=\{\mathbf{𝐯}\mid B(\mathbf{𝐯},\mathbf{𝐰})=0\text{ for all }\mathbf{𝐰}\in W\}.$$

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, $$B(\mathbf{𝐮},\mathbf{𝐯})\le C\Vert \mathbf{𝐮}\Vert \Vert \mathbf{𝐯}\Vert .$$

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, $$B(\mathbf{𝐮},\mathbf{𝐮})\ge c{\Vert \mathbf{𝐮}\Vert }^2.$$

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 Template:Mvar and linear maps V ⊗ V → K. If B is a bilinear form on Template:Mvar the corresponding linear map is given by Template:Block indent 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 Template:Mvar 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 char(K) ≠ 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 Template:Block indent

Here we still have induced linear mappings from Template:Mvar to W^∗, and from Template:Mvar 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".Template:Sfn 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 Template:Mvar, the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form $$\sum _{k=1}^p{x}_k{y}_k-\sum _{k=p+1}^n{x}_k{y}_k$$ is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:Template:Sfn Template:Quote

General modules

Given a ring Template:Mvar and a right [[Module (mathematics)|Template:Mvar-module]] M and its dual module M^∗, a mapping B : M^∗ × M → R is called a bilinear form if Template:Block indent Template:Block indent Template:Block indent 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.Template:Sfn

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.

References

• . Also: Template:Trim&pg=PA390 Bilinear form, p. 390, at Google Books

External links

• Template:Springer
• Template:Planetmath reference

Template:Functional Analysis This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.