**NONNEGATIVE MATRIX
THEORY:**

**GENERALIZATIONS AND
APPLICATIONS**

**A
Workshop held at the**

American
Institute of Mathematics

**Judi
McDonald**,

**Hans
Schneider**, University of
Wisconsin-Madison

**Michael Tsatsomeros**,

· 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)=e^{2πi/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.