Nonnegative Matrix Theory: Generalizations and Applications

December 1 to December 5, 2008

at the

American Institute of Mathematics, San Jose, California

organized by

Judith McDonald, Hans Schneider, and Michael Tsatsomeros

Original Announcement

This workshop will be devoted to the study of nonnegative matrices and their generalizations.

Nonnegative matrix theory is the study of matrices whose entries are nonnegative numbers. It is an important area of mathematics that has been built up from the illustrious Perron-Frobenius Theorem and has largely been driven by applications. Generalizations of nonnegative matrix theory typically fall into two related categories: Studying operators with Perron-Frobenius properties in various algebraic settings, and generalizing entrywise nonnegativity to other types of nonnegativity, e.g., with respect to a convex cone. This workshop will bring together individuals with experience and interests in classical nonnegative matrix theory, as well as in a variety of generalizations and applications. Specifically, the workshop will focus on the following areas:

  1. Spectral properties of nonnegative matrices. Of particular interest is the peripheral spectrum of a nonnegative matrix and the associated eigenspaces.
  2. The inverse eigenvalue problem for nonnegative matrices. Review, apply and extend recent techniques and developments toward a solution to this fundamental problem.
  3. Eventually nonnegative matrices. Obtain practical characterizations and pursue a theoretical analysis applicable to the study of positive linear (control) systems.
  4. Nonnegative matrices over cones. Examine the role and consequences of the above issues to the theory and applications of cone nonnegativity.
  5. Matrices in the Max Algebra. Develop a comprehensive Perron-Frobenius theory of nonnegative matrices under max algebra rules and, in particular, study combinatorial aspects of the associated Perron eigenspaces.
Our goal is to make progress both in specific areas and on the global themes that bring these areas together.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Paths of matrices with the strong Perron-Frobenius property converging to a given matrix with the Perron-Frobenius property
by  Abed Elhashash, Uriel G. Rothblum and Daniel B. Szyld
Sign patterns that require or allow power-positivity
by  M. Catral, L. Hogben, D. Olesky, P. van den Driessche
Sign patterns that allow eventual positivity
by  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 require eventual positivity or require eventual nonnegativity
by  Elisabeth M. Ellison, Leslie Hogben, and Michael J. Tsatsomeros