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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form (also informally a two-form) on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K which is linear in each argument separately:

B(u + v, w) = B(u, w) + B(v, w)
B(u, v + w) = B(u, v) + B(u, w)
B(λu, v) = B(u, λv) = λB(u, v)

The definition of a bilinear form can be extended to include modules over a commutative 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 ≅ Kⁿ be an n-dimensional vector space with basis {e₁, ..., e_n}. Define the n × n matrix A by A_ij = B(e_i, e_j). If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

\( B(\mathbf{v}, \mathbf{w}) = x^\mathrm T Ay = \sum_{i,j=1}^n a_{ij} x_i y_j. \)

Suppose {f1, ..., fn} is another basis for V, such that:

[f1, ..., fn] = [e1, ..., en]S

where S ∈ GL(n, K). Now the new matrix representation for the bilinear form is given by: S^TAS.

Maps to the dual space

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.

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:

\( B(x,y)=0\, \) 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 ring is that a bilinear form is unimodular if V → V^∗ is an isomorphism. Given a finite-dimensional 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

B∗(v, w) = B(w, v).

The left radical and right radical of the form B are the kernels of B₁ and B₂ respectively;^[1] they are the vectors orthogonal to the whole space on the left and on the right.^[2]

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 and only 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 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 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 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 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. If, however, char(K) = 2 then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms which are not 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(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

\(B^{\pm} = \frac{1}{2} (B \pm B^*) \)

where B∗ is the transpose of B (defined above).
Derived 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.

Reflexivity and orthogonality

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 and only if B(v, w) = 0.

A form B is reflexive if and only if it is either symmetric or alternating.^[3] 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^[4]

\(W^{\perp}=\{\mathbf{v} \mid B(\mathbf{v}, \mathbf{w})=0\ \forall \mathbf{w}\in W\} \ . \)

For a non-degenerate form on a finite dimensional space, the map W ↦ W^⊥ is bijective, and the dimension of W^⊥ is dim(V) − dim(W).

Different spaces

Much of the theory is available for a bilinear mapping to the base field

B : V × W → K.

In this situation 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".^[5] 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

\(\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:[6]

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.

Relation to tensor products

By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with 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)

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)∗ ≅ V∗ ⊗ V∗

Likewise, symmetric bilinear forms may be thought of as elements of Sym²(V^∗) (the second symmetric power of V^∗), and alternating bilinear forms as elements of Λ²V^∗ (the second exterior power of V^∗).

On normed vector spaces

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,

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

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,

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

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Bilinear form