Constructing a bilinear form on $\mathbb{R}^2$ that gives rise to a particular matrix As the title says, I'm trying to create a bilinear form $B(\cdot, \cdot)$ on $\mathbb{R}^2$ with some particular constraints (which I do not know as yet) related to the Lorentzian space $\mathbb{R}^{1, 1}$ that will somehow give rise to the matrix representation $$ A= \begin{pmatrix} 1 & 0 \\\ 0 & -1 \\\ \end{pmatrix}$$ of signature $+-$. Wouldn't the appropriate bilinear form just be the Lorentz inner product on $\mathbb{R}^2$? The Lorentz inner product on $\mathbb{R}^4$ is related to special relativity, where with the metric tensor induced by this Lorentz inner product $\mathbb{R}^4$ is now viewed as the Minkowski space $\mathbb{R}^{3,1}$.

Couldn't you just define $\langle \cdot, \cdot \rangle : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ by $\langle x, y\rangle = x_1 y_1 - x_2y_2$?

**Linearity in first entry:**

Let $c \in \mathbb{R}$ and $x,y,z \in \mathbb{R}^2$, then

\begin{eqnarray*} \langle cx + y, z\rangle & = & (cx_1+y_1)z_1 - (cx_2 + y_2)z_2 \\\ & = & c(x_1z_1 - x_2z_2) + (y_1z_1 - y_2z_2) \\\ & = & c \langle x,z\rangle + \langle y,z\rangle. \end{eqnarray*}

**Symmetry:**

$$ \langle x,y\rangle \;\; =\;\; x_1y_1 - x_2y_2 \;\; =\;\; y_1x_1 - y_2x_2 \;\; =\;\; \langle y,x\rangle. $$

This demonstrates bilinearity, and I don't think you need positive-definiteness since this is a Lorentz metric.