Nonnegative Matrix Theory: Generalizations and Applications
December 1 to December 5, 2008
American Institute of Mathematics,
San Jose, California
and Michael Tsatsomeros
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:
Our goal is to make progress both in specific areas and on the global themes that bring these
- Spectral properties of nonnegative matrices. Of particular interest is the peripheral
spectrum of a nonnegative matrix and the associated eigenspaces.
- The inverse eigenvalue problem for nonnegative matrices. Review, apply and extend recent
techniques and developments toward a solution to this fundamental problem.
- Eventually nonnegative matrices. Obtain practical characterizations and pursue a
theoretical analysis applicable to the study of positive linear (control) systems.
- Nonnegative matrices over cones. Examine the role and consequences of the above issues
to the theory and applications of cone nonnegativity.
- 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.
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