Given a matrix of a bilinear form, how do we find the explicit bilinear form function?

Question

Say, we are given the matrix of a bilinear form: $\begin{bmatrix} 1 & 0\\ 0 &-1 \end{bmatrix}$.

How do we find the explicit bilinear form from where this matrix came from?

Answer 1

Given a square matrix $A$ and input vectors $x$ and $y$, you get a bilinear form by computing $x^TAy$.

In your example, $$\left[\begin{matrix} x_1 & x_2\end{matrix}\right]\left[\begin{matrix} 1 & 0\\ 0 & -1\end{matrix}\right]\left[\begin{matrix} y_1 \\ y_2\end{matrix}\right] = \left[\begin{matrix} x_1 & x_2\end{matrix}\right]\left[\begin{matrix}y_1\\ -y_2\end{matrix}\right] = x_1y_1 - x_2y_2$$

In the general $2\times 2$ case, $$\left[\begin{matrix} x_1 & x_2\end{matrix}\right]\left[\begin{matrix} a & b\\ c & d\end{matrix}\right]\left[\begin{matrix} y_1 \\ y_2\end{matrix}\right] = \left[\begin{matrix} x_1 & x_2\end{matrix}\right]\left[\begin{matrix}ay_1+by_2\\ cy_1 + dy_2\end{matrix}\right] = ax_1y_1 + bx_1y_2 + cx_2y_1 + dx_2y_2$$