Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

TL;DR

This paper characterizes the semigroup of ratios of products of principal minors of positive definite matrices, providing a complete description for 4x4 matrices and conjecturing a finite characterization for larger sizes.

Contribution

It introduces a cone-theoretic approach to determine the bounded ratios semigroup, identifying new generators beyond classical inequalities for 4x4 matrices.

Findings

Complete semigroup description for 4x4 matrices

Identification of new generators not implied by Hadamard-Fischer inequalities

Conjecture on finite determination of bounded ratios for larger matrices

Abstract

Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Hadamard-Fischer and ratios not bounded by 1. For n > 4 it is shown that the containment of semigroups is strict, but a generalization of nullity patterns, of which one example is given, is conjectured to provide a finite determination of all bounded ratios.