1.2.25 Bilinear Form Definition

A bilinear form is a function that takes two vectors and returns a scalar, linear in each input, foundational in algebra and tensor theory.

Bilinear Form Definition is the characterization of a function that takes two vector arguments, possibly from different vector spaces, and returns a scalar, while being linear in each of its two arguments separately when the other argument is held fixed. A bilinear form is the simplest nontrivial case of a multilinear form and generalizes familiar constructions such as the dot product, inner products, and the pairing between a vector space and its dual.

Formal Definition

Let $V$ and $W$ be vector spaces over a common field $F$ . A function

B : V \times W \to F

is called a bilinear form if it satisfies the following two conditions for all $u, v \in V$ , all $w, x \in W$ , and all scalars $a, b \in F$ .

Linearity in the first argument:

B (a u + b v, w) = a B (u, w) + b B (v, w)

Linearity in the second argument:

B (u, a w + b x) = a B (u, w) + b B (u, x)

Both conditions must hold independently; linearity in one argument does not imply linearity in the other, so a genuine bilinear form must be checked, and must hold, in each slot separately. When $V = W$ , the bilinear form is said to be defined on $V$ alone, and the two arguments are drawn from the same space.

Basic Consequences of the Definition

Directly from bilinearity, a bilinear form vanishes whenever either argument is the zero vector,

B (0, w) = B (v, 0) = 0

and it distributes fully over sums in both arguments simultaneously, so that for finite sums $v = \sum_{i} a_{i} v_{i}$ and $w = \sum_{j} b_{j} w_{j}$ ,

B (v, w) = \sum_{i, j} a_{i} b_{j} B (v_{i}, w_{j})

This double-sum expansion is what allows a bilinear form on finite-dimensional spaces to be fully reconstructed from its values on basis vectors alone.

Classification on a Single Vector Space

When a bilinear form is defined on a single vector space $V$ , it can be classified according to its behavior under exchange of its two arguments.

Symmetric Bilinear Forms

A bilinear form $B$ is symmetric if

B (v, w) = B (w, v)

for all $v, w \in V$ . Symmetric bilinear forms are the algebraic objects underlying inner products and quadratic forms, since every symmetric bilinear form gives rise to a quadratic form through $Q (v) = B (v, v)$ .

Antisymmetric (Alternating) Bilinear Forms

A bilinear form is antisymmetric, or alternating, if

B (v, w) = - B (w, v)

for all $v, w \in V$ , which in particular forces $B (v, v) = 0$ for every $v$ outside fields of characteristic two. Antisymmetric bilinear forms underlie symplectic geometry, where a nondegenerate antisymmetric bilinear form on a vector space defines a symplectic structure.

General Bilinear Forms

A bilinear form need not be symmetric or antisymmetric. Over a field of characteristic other than two, any bilinear form $B$ decomposes uniquely into a symmetric part and an antisymmetric part,

B (v, w) = \frac{B (v, w) + B (w, v)}{2} + \frac{B (v, w) - B (w, v)}{2}

confirming that the symmetric and antisymmetric families together account for the full space of bilinear forms.

Matrix Representation

On finite-dimensional spaces, a bilinear form is completely determined by a matrix of its values on basis vectors. If ${e_{1}, \dots, e_{n}}$ is a basis of $V$ and ${f_{1}, \dots, f_{m}}$ is a basis of $W$ , the entries

M_{i j} = B (e_{i}, f_{j})

define an $n \times m$ matrix $M$ , and for any vectors $v$ and $w$ with coordinate column vectors $[v]$ and $[w]$ relative to these bases,

B (v, w) = [^{T} v] M [w]

A symmetric bilinear form corresponds to a symmetric matrix, an antisymmetric bilinear form corresponds to a skew-symmetric matrix, and a change of basis transforms the representing matrix by congruence rather than by the similarity transformation used for linear operators, reflecting the fact that a bilinear form is a different kind of object from a linear map.

Nondegeneracy

A bilinear form $B$ on $V \times W$ is nondegenerate if the only vector $v$ satisfying $B (v, w) = 0$ for every $w \in W$ is the zero vector, and symmetrically for $W$ . Nondegenerate bilinear forms establish an isomorphism between a vector space and the dual of the other, which is the mechanism by which an inner product identifies a finite-dimensional space with its own dual space. Degenerate bilinear forms, by contrast, possess a nonzero radical — the set of vectors annihilated by the form — and fail to induce such an identification.

Relationship to Tensors

A bilinear form is precisely a covariant tensor of order two: an element of the space of multilinear maps taking two vector arguments and returning a scalar. This identification places bilinear forms as the degree-two instance of the general hierarchy of multilinear forms, sitting between linear forms of degree one and higher-order multilinear forms of degree three and above. Within tensor algebra, bilinear forms provide the simplest nontrivial setting in which the distinction between symmetric and antisymmetric tensor structure, matrix representation under change of basis, and the notion of nondegeneracy can all be studied before being generalized to tensors of arbitrary order.