Bilinear map

A bilinear map is a function f: V × W → Z, where V, W, and Z are vector spaces over the same field K, such that for each fixed w the map v ↦ f(v, w) is linear, and for each fixed v the map w ↦ f(v, w) is linear.¹ This means f(av + bv′, w) = af(v, w) + bf(v′, w) and similarly in the second argument. Bilinearity is distinct from linearity of the combined map; for example, the dot product in Rⁿ is bilinear but not linear as a function Rⁿ × Rⁿ → R because f(λv, λw) = λ²f(v, w).

Bilinear maps are ubiquitous. The multiplication of scalars in a field is bilinear, as is matrix multiplication when viewed as a map from M_m×n × M_n×p → M_m×p.² In geometry, the inner product and the cross product (in R³) are bilinear. In analysis, the convolution of functions is bilinear, and the integral of a product of two functions defines a bilinear pairing. In algebra, the tensor product is defined by a universal property that characterizes bilinear maps: every bilinear map V × W → Z factors uniquely through the tensor product V ⊗ W.

A bilinear form is a bilinear map V × V → K. Over finite-dimensional vector spaces, every bilinear form can be represented by a matrix A such that f(v, w) = v^TAw.³ Symmetric bilinear forms correspond to symmetric matrices and are central to the study of quadratic forms and geometry (e.g., the metric tensor in relativity). Alternating bilinear forms, where f(v, v) = 0, correspond to skew-symmetric matrices and are fundamental in symplectic geometry and the theory of determinants.

Bilinear maps extend beyond vector spaces to modules over rings, where the scalar action may differ on each side, giving rise to balanced maps.⁴ In functional analysis, a continuous bilinear map between Banach spaces is bounded if and only if it is continuous at the origin, a result that does not hold for general multilinear maps. In cryptography, bilinear maps (pairings) on elliptic curves, such as the Weil and Tate pairings, enable advanced protocols like identity-based encryption and short signatures.⁵ The term 'bilinear' also appears in approximation theory, where bilinear interpolation uses a bilinear function to estimate values on a grid.

Tensor product

A construction that turns bilinear maps into linear maps on a new vector space.

Bilinear form

A bilinear map from V × V to the base field.

Pairing

A bilinear map used in cryptography, often denoted e: G1 × G2 → GT.