Optimizing and Factorizing the Wilson Matrix

Abstract

The Wilson matrix, $W$, is a $4\times 4$ unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940s, owing to its mild ill-conditioning. We ask how close $W$ is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations $W = Z^TZ$ with $Z$ having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that $W$ has an integer factor $Z$ and two rational factors, up to signed permutations. This little $4 \times 4$ matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results