From b71651d901d318a0c827bcbab4d2fcdfdf282ea9 Mon Sep 17 00:00:00 2001 From: Glen Whitney Date: Wed, 19 Feb 2025 20:46:41 +0000 Subject: [PATCH] Update Gram matrix parameterization --- Gram-matrix-parameterization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Gram-matrix-parameterization.md b/Gram-matrix-parameterization.md index e5da45d..2db9dee 100644 --- a/Gram-matrix-parameterization.md +++ b/Gram-matrix-parameterization.md @@ -148,7 +148,7 @@ Let $A_K \colon \R^K \to V$ be the restriction of $A$. The first five rows of $U 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. Let's treat some of the unknown entries as independent variables, and the rest as dependent variables. Whenever we find a six-by-six submatrix where one entry is dependent and the rest are known or independent, knowing that the corresponding minor is zero gives us a quadratic equation for the dependent variable. Treating that dependent variable as "1-knowable" may provide further six-by-six submatrices where one entry is dependent and the rest are known, independent, or 1-knowable. The resulting quadratic equation for the dependent variable makes that variable "2-knowable", and the process continues. -For $G$ to be realizable as a Gram matrix, it's necessary for each dependent variable to be real. Whenever we have a quadratic equation for a dependent variable, the discriminant of the equation gives us a consistency constraint inequality. The complexity of the consistency constraint for an $n$ - knowable variable gets more complicated as $n$ increases. +For $G$ to be realizable as a Gram matrix, it's necessary for each dependent variable to be real. Whenever we have a quadratic equation for a dependent variable, the discriminant of the equation gives us a consistency constraint inequality. The complexity of the consistency constraint for an $n$-knowable variable gets more complicated as $n$ increases. ### Choosing dependent variables