bilinear map

A multilinear map with two arguments. A function $f:V\times W\to X$ that is linear in each argument separately:
$f(\alpha u_1+u_2,w)=\alpha f(u_1,w)+f(u_2,w)$
$f(u,\alpha w_1+w_2)=\alpha f(u,w_1)+f(u,w_2)$

The dot product $⟨u,v⟩=u^{\top }v$ is the simplest example.

Concretely: a bilinear map can weight products of pairs $u_i\cdot w_j$ differently, but that’s all it can do. Hold one input fixed and the output changes linearly with the other. It can ask “how much of $u_3$ times how much of $w_5$” but never “is $u_3$ bigger than $w_5$” or “are $u_3$ and $w_5$ both large.”

Bilinear $\ne$ linear transformation

$f(\alpha u,\alpha w)={\alpha }^{2}f(u,w)$, not $\alpha f(u,w)$. Linearity in each argument composes multiplicatively, giving quadratic scaling in the pair.

Bilinear forms as matrices

Any bilinear form $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ can be written as $f(u,w)=u^{\top }Mw$ for some matrix $M\in {\mathbb{R}}^{n\times m}$.
The entries $M_{ij}$ are the coefficients of pairwise products $u_i{w}_j$.
→ bilinear maps can only capture pairwise multiplicative interactions between coordinates of $u$ and $w$.