Matrices totally positive relative to a tree

Date: 2016..12..15
Event: Gamma seminar
Venue: Room 2.2.D08, Math department. Universidad Carlos III de Madrid

Abstract

A matrix is called totally positive (TP) if every minor of it is positive. It is known that for a 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. We will be interested in submatrices of a given matrix that are TP, or permutation similar to TP.

Thus, we will be interested in permuted submatrices, identified by ordered index lists. For a given labelled tree T on n vertices, we say that A is T-TP if, for every path P in T, A[P] is TP. J. Garloff relayed to us an old conjecture of A. Neumaier that for any tree T, the eigenvector associated with the smallest eigenvalue of a T-TP matrix should be signed according to the labelled tree T. We refer to this as "the T-TP conjecture".

In this talk we prove that the conjecture is false, giving some examples and present. certain weakening of the TP hypothesis is shown to yield a similar conclusion, i.e. the eigenvector, associated with the smallest eigenvalue, alternates in sign as in the tree. This is a joint work with Charles R. Johnson, and Boris Tadchiev.

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