Bilinear form

In, a bilinear form on a V is a  V × V → K, where K is the  of s.  In other words, a bilinear form is a function B : V × V → K that is  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 definition of a bilinear form can be extended to include over a, with s replaced by s.

When K is the field of s C, one is often more interested in s, which are similar to bilinear forms but are in one argument.

Coordinate representation
Let $V ≅ K^{n}$ be an $n$-dimensional vector space with basis ${e_{1}, ..., e_{n}}.|undefined$

The n × n matrix A, defined by Aij = B(ei, ej) is called the matrix of the bilinear form on the basis ${e_{1}, ..., e_{n}}.|undefined$

If the $n × 1$ matrix $x$ represents a vector $v$ with respect to this basis, and analogously, $y$ represents another vector $w$, then:


 * $$B(\mathbf{v}, \mathbf{w}) = \mathbf{x}^\textsf{T} A\mathbf{y} = \sum_{i,j=1}^n x_i a_{ij} y_j. $$

A bilinear form has different matrices on different bases. However, the matrices of a bilinear on different bases are all. More precisely, if ${f_{1}, ..., f_{n}}|undefined$ is another basis of $V$, then
 * $$\mathbf{f}_j=\sum_{i=1}^n S_{i,j}\mathbf{e}_i,$$

where the $$S_{i,j}$$ form an $S$. Then, the matrix of the bilinear form on the new basis is $S^{T}AS$.

Maps to the dual space
Every bilinear form B on V defines a pair of linear maps from V to its 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 is to be placed (see ).

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. 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  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 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 s 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 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, 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 then, relative to some  for V, a bilinear form is degenerate if and only if the  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 ). 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 (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 bilinear form to be
 *  if B(v, w) = B(w, v) for all v, w in V;
 *  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 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 that are not alternating.

A bilinear form is symmetric (resp. skew-symmetric) its coordinate matrix (relative to any basis) is  (resp. ). 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
 * $$B^{+} = \tfrac{1}{2} (B + {}^{\text{t}}B) \qquad  B^{-} = \tfrac{1}{2} (B - {}^{\text{t}}B) ,$$

where tB is the transpose of B (defined above).

Derived quadratic form
For any bilinear form B : V × V → K, there exists an associated 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 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 ⇔ 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 


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

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

Different spaces
Much of the theory is available for a 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 Aij having only +1 or −1 for non-zero elements. Some of the "inner products" are and some are s or. Rather than a general field K, the instances with real numbers R, complex numbers C, and 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:
 * 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  or Minkowski spacetime. The special case R(p, p) will be referred to as the split-case.

Relation to tensor products
By the of the, 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 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 Sym2(V∗) (the second of V∗), and alternating bilinear forms as elements of Λ2V∗ (the second  of V∗).

On normed vector spaces
Definition: A bilinear form on a  (V, ‖·‖) is bounded, if there is a constant C such that for all u, v ∈ V,
 * $$ B ( \mathbf{u}, \mathbf{v}) \le C \left\| \mathbf{u} \right\| \left\|\mathbf{v} \right\| .$$

Definition: A bilinear form on a normed vector space (V, ‖·‖) is elliptic, or, if there is a constant c > 0 such that for all u ∈ V,
 * $$ B ( \mathbf{u}, \mathbf{u}) \ge c \left\| \mathbf{u} \right\| ^2 .$$

Generalization to modules
Given a R and a right  M and its  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∗, x, y ∈ M, α, β ∈ R.

The mapping ⟨&sdot;,&sdot;⟩ : M∗ × M → R : (u, x) ↦ u(x) is known as the , 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 of M.