Two absolutely bounded determinantal ratios

Abstract

Bounded ratios of products of minors of positive definite matrices have a long history, starting with Hadamard's inequality in 1893. It states that for every positive semidefinite matrix $A$ $$\det A\le A_{11}\cdots A_{nn}.$$ This inequality was subsequently generalized by Fisher and then further by Koteljanskii. The latter states that for every positive semidefinite matrix $A$ and any index sets ${\alpha }_1,{\alpha }_2\subseteq \{1,\dots ,n\}$ one has $$\det A[{\alpha }_1\cup {\alpha }_2]\det A[{\alpha }_1\cap {\alpha }_2]\le \det A[{\alpha }_1]\det A[{\alpha }_2],$$ where $A[\alpha ]$ denotes the principal submatrix determined by the indexes in $\alpha$. In a manuscript published only on the arXiv in 2008, Hall and Johnson made three conjectures about ratios of products of principal minors of $4\times 4$ positive definite matrices, denoted by $R_i$, $i=1,2,3$, see (2) and (3). They hypothesized that the supremum of $R_1$ was $27/16$, while the supremum of the other two ratios was $1$. Such ratios are called absolutely bounded. The conjecture for $R_1$ was affirmed in [17] and it is the only known bounded determinantal ratio with supremum bigger than one. The goal of this paper is to affirm the conjecture for $R_2$ and $R_3$. It is known that the upper bound for the ratios $R_i$, $i=1,2,3$, does not follow from repeated applications of Koteljanskii's inequality. In addition, Hall and Johnson showed that $R_i$ is bounded above by $4$, for $i=1,2,3$.

14 pages