NONNEGATIVE MATRIX THEORY:

GENERALIZATIONS AND APPLICATIONS

A Workshop held at the

American Institute of Mathematics

Palo Alto, December 1-5, 2008

Judi McDonald, Washington State University

Hans Schneider, University of Wisconsin-Madison

Michael Tsatsomeros, Washington State University

 

·        A special volume of the Electronic Journal of Linear Algebra (ELA) will be devoted to the workshop. It will contain papers on work presented, considered or inspired by the themes of the workshop.

·        AIM will maintain a web page dedicated to the workshop. It includes links to the schedule, the participant list, etc.

·        The organizers filed this report on the workshop activities.

·        Here are the slides of some of the main talks given at the workshop:

Spectral Properties of Nonnegative Matrices by Danny Hershkowitz.

Max-algebra: Achievements and Challenges by Peter Butkovic.

The Nonnegative Inverse Eigenvalue Problem by Tom Laffey.

Nonnegative Linear Algebra and Max Linear Algebra: Where's the Difference? by Hans Schneider.


Broadly stated, our goals at the workshop included:

·         making progress on important problems in classical nonnegative matrix theory

·         further developing other types of nonnegativity in a linear algebraic setting

·         highlighting and contributing to the solution of questions

·         relating to nonnegativity that arise in applications

·         finding global themes that tie notions of nonnegativity together

Below are some write-ups and discussion that led up to the workshop.

For a list of standard notations and definitions see Standard Notation and Definitions. The foundations for nonnegative matrix theory trace back to the works of Perron, and later Frobenius. Their work has led to the discovery of many interesting properties of nonnegative irreducible matrices:

Theorem [Perron--Frobenius Theorem and extensions]

Let A be an nxn entrywise nonnegative irreducible matrix. Then there exists a positive integer h such that

a)      ρ(A) is a simple eigenvalue of A,

b)      there exists a positive vector x such that Ax= ρ(A)x,

c)      π(A)= a complete set of h-th roots of unity multiplied by ρ(A),

d)     σ(A)=e2πi/h σ(A),

e)      h is equal to the greatest common divisor of the cycle lengths in the digraph of A.

These ideas have been extended in many different interesting directions (click on each title for more information):

The Inverse Eigenvalue Problem

Spectral Properties of Reducible Nonnegative and Eventually Nonnegative Matrices

Eventually, Essentially and Exponentially Nonnegative Matrices

Properties of Nonnegative Matrices in the Max Algebra

Matrices that Share the Properties of Nonnegative Matrices

Matrices that Leave a Cone Invariant

Applications of Nonnegative Matrices


For questions, comments or to add material to this site, contact Michael Tsatsomeros.