# Eventually Nonnegative Matrices and their Sign Patterns (11frg149)

## Organizers

Minerva Catral (Xavier University)

Craig Erickson (Iowa State University)

Leslie Hogben (Iowa State University)

Dale Olesky (University of Victoria)

Pauline van den Driessche (University of Victoria)

## Objectives

The objective of the proposed Focused Research Group is the study of sub-classes of the eventually nonnegative matrices for which aspects of the Perron-Frobenius theory hold. Relationships between the sign pattern classes of such matrices and of related classes are of particular interest. The concept of strong eventual nonnegativity was introduced in [3], specifying a subset of the eventually nonnegative matrices that excludes nilpotent matrices and for which Perron-Frobenius theorems hold that are similar to those for eventually positive matrices.

The study of sign patterns that allow strong eventual nonnegativity has only just begun and is one of the questions to be investigated. (It is a consequence of [7, Corollary 2.2] that a sign pattern requires strong eventual nonnegativity if and only if it is irreducible and nonnegative.) Another related question that was raised at the recent Western Canada Linear Algebra Meeting (WCLAM) 2010 is to determine what sign patterns allow a power that is a stochastic matrix.

In the paper [4], it was shown that if $A$ is a strongly eventually nonnegative matrix such that rank $A^2$ = rank $A$ and $A$ has $r$ dominant eigenvalues, then $A$ is $r$-cyclic. Further investigation of eventually nonnegative matrices such that rank $A^2$ = rank $A$ seems warranted and is related to an open question of Zaslavsky and Tam [9, Question 6.3], as well as the results in [5].

This group of 5 researchers has been working together for 9 months and grew out of an American Institute of Mathematics workshop, ``Nonnegative Matrix Theory: Generalizations and Applications," held in December 2008. At that workshop, Catral, Hogben, Olesky, and van den Driessche began working together within a larger group. This collaboration continued with a visit by Hogben to the University of Victoria (where Catral was then a post-doc) in March 2009. Hogben's doctoral student Erickson joined the group in Fall 2009 (when Catral also arrived at ISU as Hogben's post-doc) and the idea of strongly eventually nonnegative matrices was developed during a visit by Olesky to Iowa State University in November 2009. The group has worked in two subgroups, one at University of Victoria and one at Iowa State University, but as of Fall 2010, Catral will be at Xavier University, so the group will be spread across three locations (and Catral will be isolated). Assorted subgroups have gathered briefly at various places including the WCLAM held at BIRS in May 2010 (Catral, Hogben, Olesky, van den Driessche), but the group needs time to focus on the project with the whole group present and without other distractions. BIRS is an ideal setting not only because the 5 group members need to be physically in the same place, but precisely because all members will be away from the distractions of their home institutions and because Banff is centrally located from the three universities.

Papers already produced as part of this collaboration:

1. Abraham Berman, Minerva Catral, Luz M. Dealba, Abed Elhashash, Frank J. Hall, Leslie Hogben, In-Jae Kim, D. D. Olesky, Pablo Tarazaga, Michael J.

Tsatsomeros, P. van den Driessche. Sign patterns that allow eventual positivity. Electronic Journal of Linear Algebra 19 (2010) 108-120.

2. M. Catral, L. Hogben, D. D. Olesky, P. van den Driessche. Sign patterns that require or allow power-positivity. Electronic Journal of Linear Algebra 19 (2010) 121-128.

3. M. Catral, C. Erickson, L. Hogben, D. D. Olesky, P. van den Driessche. Eventually nonnegative matrices and related classes. Under review.

4. L. Hogben. Determining whether a matrix is strongly eventually nonnegative. Under review.

Other papers cited:

5. S. Carnochan Naqvi and J. J. McDonald. The combinatorial structure of eventually nonnegative matrices. Electronic Journal of Linear Algebra 9 (2002): 255-269.

6. S. Friedland, On an inverse problem for nonnegative and eventually nonnegative matrices, Israel Journal of Mathematics, 29 (1978): 43-60.

7. E. M. Ellison, L. Hogben, M. J. Tsatsomeros. Sign patterns that require eventual positivity or require eventual nonnegativity. Electronic Journal of Linear Algebra, 19 (2010): 98-107.

8. D. Handelman, Positive matrices and dimension groups affiliated to $C^*$-algebras and topological Markov chains. Journal of Operator Theory 6 (1981): 55-74.

9. B. G. Zaslavsky and B.-S. Tam. On the Jordan form of an irreducible matrix with eventually non-negative powers. Linear Algebra and its Applications, 302-303 (1999): 303-330.