**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.

## Download

Link | Size | Description |
---|---|---|

seminar_22.pdf | 716 KB | Slides (PDF, 19 pages) |