Bilinear Forms

Let $n$ be a positive integer. A function $B:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ is said to be a bilinear form if there is an $n\times n$ matrix $A$ such that

$$B(\mathbf{x},\mathbf{y})={\mathbf{x}}^{T}A\mathbf{y},\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$$

The function $B$ is linear in argument separately, which is why we call them bilinear.

For $\mathbf{y}$ fixed, all $\mathbf{u},\mathbf{v}\in {\mathbb{R}}^n$ , and any real constant $c$, linearity in the first argument means the following:

$$\begin{array}{rl}B(\mathbf{u}+\mathbf{v},\mathbf{y})& =B(\mathbf{u},\mathbf{y})+B(\mathbf{v},\mathbf{y})\\ B(c\mathbf{u},\mathbf{y})& =cB(\mathbf{u},\mathbf{y})\end{array}$$

For $\mathbf{x}$ fixed, all $\mathbf{w},\mathbf{z}\in {\mathbb{R}}^n$, and any real constant $c$, linearity in the second argument means the following:

$$\begin{array}{rl}B(\mathbf{x},\mathbf{w}+\mathbf{z})& =B(\mathbf{x},\mathbf{w})+B(\mathbf{x},\mathbf{z})\\ B(\mathbf{x},c\mathbf{w})& =cB(\mathbf{x},\mathbf{w})\end{array}$$