Matrices Totally Positive Relative to a Tree
Charles R. Johnson, Roberto S. Costas-Santos, and Boris Tadchiev

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.
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