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Update Gram matrix parameterization

Glen Whitney 2025-02-19 20:46:41 +00:00
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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. 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 ### Choosing dependent variables