Bilinear form

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

B₁(v)(w) = B(v, w) B₂(v)(w) = B(w, v)

This is often denoted as

B₁(v) = B(v, ⋅) B₂(v) = B(⋅, v)

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:

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

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

The corresponding notion for a module over a commutative ring is that a bilinear form is unimodular 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

^tB(v, w) = B(w, v).

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:

Definition:B is nondegenerate if B(v, w) = 0 for all w implies v = 0.

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

B(v, w) = A(v)(w).

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

• symmetric if B(v, w) = B(w, v) for all v, w in V;
• alternating if B(v, v) = 0 for all v in V;
• skew-symmetric or antisymmetric if B(v, w) = −B(w, v) for all v, w in V;

  Proposition

  Every alternating form is skew-symmetric.

  Proof

  This can be seen by expanding 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 $${\displaystyle B^{+}=\frac{1}{2}(B+{}^{\text{t}}B)B^{-}=\frac{1}{2}(B-{}^{\text{t}}B),}$$ where ^tB is the transpose of B (defined above).

Reflexive bilinear forms and orthogonal vectors

Definition: A bilinear form B : V × V → K is called reflexive if B(v, w) = 0 implies B(w, v) = 0 for all v, w in V. Definition: Let B : V × V → K be a reflexive bilinear form. v, w in V are orthogonal with respect to B if B(v, w) = 0.

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 ⇔ 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 W is a subspace. Define the orthogonal complement $${\displaystyle W^{\perp }=\left\{\mathbf{v}\mid B(\mathbf{v},\mathbf{w})=0\text{ for all }\mathbf{w}\in W\right\}.}$$

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

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

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 V and linear maps V ⊗ V → K. If B is a bilinear form on V the corresponding linear map is given by

v ⊗ w ↦ B(v, w)

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 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

B : V × W → K.

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 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 K, the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form $${\displaystyle \sum _{k=1}^p{x}_k{y}_k-\sum _{k=p+1}^n{x}_k{y}_k}$$ is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:

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

B(u + v, x) = B(u, x) + B(v, x) B(u, x + y) = B(u, x) + B(u, y) B(αu, xβ) = αB(u, x)β

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.