Observe that the Gram matrix represents the pullback of the Lorentz form

Gram-matrix-parameterization.md

@@ -6,6 +6,11 @@ In [inversive coordinates](https://code.studioinfinity.org/glen/dyna3/src/branch
 
 ### Constraints as Gram matrix entries
 
-The vectors $v_1, \ldots, v_n \in \mathbb{R}^{1,4}$ representing the elements of our construction can be encoded in a linear map $\mathbb{R}^n \to \mathbb{R}^{1,4}$, whose matrix is
+The vectors $a_1, \ldots, a_n \in \mathbb{R}^{1,4}$ representing the elements of our construction can be encoded in a linear map $A \colon \mathbb{R}^n \to \mathbb{R}^{1,4}$, whose matrix is
-\[ V = \left[\begin{array}{cccc} \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \\ v_1 & v_2 & \ldots & v_n \\ \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \end{array}\right] \]
+\[ \left[\begin{array}{cccc} \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \\ a_1 & a_2 & \cdots & a_n \\ \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \end{array}\right] \]
-We can then express constraints by fixing elements of the Gram matrix $G = V^\top V$. The transpose is taken with respect to the Lorentz product $\langle \;\; , \;\; \rangle$, so $G_{jk} = \langle v_j, v_k \rangle$.
+We can then express constraints by fixing elements of the Gram matrix $G = A^\top A$, where $\top$ is the adjoint with respect to the inner product $\langle\_\!\_, \_\!\_\rangle$ on $\mathbb{R}^n$ and the Lorentz product $(\_\!\_, \_\!\_)$ on $\mathbb{R}^{1,4}$. The bilinear form $\langle\_\!\_, G\_\!\_\rangle$ is the pullback of the Lorentz form along $A$:
+\[\begin{align*} \langle\_\!\_, G\_\!\_\rangle & = \langle\_\!\_, A^\top A\_\!\_\rangle \\ & = (A\_\!\_, A\_\!\_). \end{align*}\]
+To confirm that $G$ is the Gram matrix of $a_1, \ldots, a_n$, observe that
+\[\begin{align*} G_{jk} & = \langle e_j, G e_k \rangle  \\  & = (A e_j, A e_k) \\ & = (a_j, a_k), \end{align*}\]
+where $e_1, \ldots, e_n$ is the standard basis for $\mathbb{R}^n$.
+