Matrices Totally Positive Relative to a Tree

TL;DR

This paper explores a weakened form of total positivity relative to a tree structure, demonstrating that similar eigenvalue and eigenvector properties hold under this relaxed condition.

Contribution

It introduces a new class of matrices with properties akin to totally positive matrices, extending known spectral results to this broader context.

Findings

Eigenvalues remain positive and distinct under the weakened condition

The eigenvector associated with the smallest eigenvalue exhibits an alternating sign pattern

The results generalize classical TP matrix properties to a tree-relative setting

Abstract

It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of the TP hypothesis is shown to yield a similar conclusion.