Open problems on GKK tau-matrices

TL;DR

The paper surveys open problems on GKK tau-matrices, a class of P-matrices that exhibit eigenvalue monotonicity and weak sign-symmetry, in light of Holtz's demonstration that some such matrices can be unstable. It situates these questions relative to known stable classes (nonsingular totally nonnegative matrices, Hermitian positive definite matrices, and M-matrices) and notes that instability occurs even within GKK tau-matrices, motivating a structured set of questions. Five core problems are outlined: closure of GKK matrices under approximation by strict GKK, stability of strict GKK and strict GKK tau-matrices, the minimal dispersal threshold for stability, the inverse-minor problem and its implications for spectra, and the validity of Newton's inequalities for GKK tau-matrices (with related subproblems). The aim is to guide future theoretical development toward concrete stability criteria for GKK tau-matrices and to illuminate implications for applications relying on stable matrix behavior.

Abstract

We propose several open problems on GKK tau-matrices raised by examples showing that some such matrices are unstable