Bilinear Functions Not Isomorphic to xy

Question

Wolfram Alpha lists $z=xy$ as an example of a bilinear function. Are there any others that aren't this form of some product of two objects?

Answer 1

Given $v,w \in \mathbb{R}^n$ (column vectors) and an $n \times n$ matrix $A$. Then $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ defined by $f(v,w)=v^tAw$ is a bilinear function.

Given a bilinear function $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$.

Fix vector $v$ and then $f(v,\cdot):\mathbb{R}^n \to \mathbb{R}$ is linear so you can find a matrix $A_v$ so that $f(v,w)=A_vw$ for all $w$ in particular this matrix has to be $1 \times n$ so it's a row-vector. Pick the standard basis for $\mathbb{R}^n$: $e_1,e_2,\ldots, e_n$ then call $A_i=A_{e_i}$ for each $i=1,\ldots,n$ and define $n \times n$ matrix $A$ with rows $A_1, \ldots, A_n$. Note that $$Aw=\begin{bmatrix}A_1w\\ A_2w\\ \vdots \\ A_nw\end{bmatrix}$$

Any $v\in \mathbb{R}^n$ has $v=\sum v_i e_i=\begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix}$ so $$v^tAw=\sum v_iA_iw=\sum v_i f(e_i,w)=\sum f(v_ie_i,w)=f\left(\sum v_i e_i,w\right)=f(v,w) $$

So all bilinear forms on a finite dimensional vector space can be defined by a matrix product.

Answer 2

Consider for any matrix $A=(a_{ij})\in M_{m,n}(K)$ a bilinear function defined by $$ f(x,y)=\sum_{i=1}^m\sum_{j=1}^na_{ij}x_ix_j. $$