Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices
2026-06-22 • Information Theory
Information Theory
AI summaryⓘ
The authors study certain ratios made from products of smaller parts (principal minors) of positive definite matrices, which are special types of matrices with nice properties. They solve a challenging optimization problem exactly, including proving a previously guessed minimum value for a specific ratio called the Ingleton ratio in 4x4 matrices. They also find that for larger matrices, the set of all these nice ratios is quite complicated and doesn't have simple geometric or algebraic descriptions. This work advances understanding of how these matrix ratios behave and their underlying structure.
positive definite matrixprincipal minormatrix inequalityoptimizationIngleton ratiopositive definite conenonconvex optimizationpolyhedral conesemialgebraic set
Authors
Tobias Boege, Ludovick Bouthat
Abstract
We study sharp inequalities for ratios of products of principal minors of real positive definite matrices. Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over $4\times 4$ positive definite matrices is $16/27$, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for $n\ge 4$, and that it is not semialgebraic over $\mathbb{Q}$.