Discuss Gram matrix consistency constraints

Gram-matrix-parameterization.md

@@ -13,8 +13,22 @@ The symmetric bilinear form $\langle\_\!\_, G\_\!\_\rangle$ is the pullback of t
 
 ## Reconstruction
 
+### Assumptions
+
+Throughout this section, we'll assume $a_1, \ldots, a_n$ span $V$, so $G$ has rank five.
+
 ### Reconstruction from the Gram matrix
 
-Let's say $a_1, \ldots, a_n$ span $V$. Write $G$ as $LU$, where the matrix $U$ is a row echelon form of $G$, and the the matrix $L$ encodes the sequence of row operations that restores $G$. Since $G$ has rank five, the first five rows of $U$ are non-zero, and the rest are zero. Thus, the first five columns of $L$ are non-zero, and the rest are zero. Since $G$ is symmetric, we should have $L = U^\top D$, where $D$ is a diagonal matrix that has one negative entry and four positive entries at the pivot positions on the diagonal.
+Write $G$ as $LU$, where the matrix $U$ is a row echelon form of $G$, and the the matrix $L$ encodes the sequence of row operations that restores $G$. Since $G$ has rank five, the first five rows of $U$ are non-zero, and the rest are zero. Thus, the first five columns of $L$ are non-zero, and the rest are zero. Since $G$ is symmetric, we should have $L = U^\top D$, where $D$ is a diagonal matrix that has one negative entry and four positive entries at the pivot positions on the diagonal.
 
 The first five rows of $U$ form the matrix of $A$ in some orthogonal basis for $V$. The matrix $D$ describes the Lorentz form in this basis.
+
+### Consistency constraints on the Gram matrix
+
+Since $G$ has rank five, every six-by-six minor is zero. Some of the entries of $G$ are known, because they're specified by constraints. Among the unknown entries, we want to treat some as independent variables, and the rest as dependent variables. We choose the independent variables so that each dependent variable is part of a six-by-six minor in which all the other entries are known or independent. Knowing that every six-by-six minor is zero then gives us a quadratic equation for each dependent variable.
+
+For $G$ to be realizable as a Gram matrix, it's necessary for each dependent variable to be real. The discriminants of the equations for the dependent variables thus give us a system of consistency constraint inequalities. To solve these inequalities, we need to distinguish between three pretty different cases.
+
+1. The solution space has non-empty interior—or, equivalently, codimension zero.
+2. The solution space has empty interior—or, equivalently, positive codimension.
+3. The solution space is empty.