Matrix Analysis

Dr. Pietro Paparella, Faculty Mentor

Title: Perron Similarities and the Nonnegative Inverse Eigenvalue Problem

Matrix Analysis is an accessible area of mathematics that has proven to be suitable and successful for undergraduate research projects (the highly sought-after Matrix Analysis REU ran at The College of William & Mary for over twenty-five years), in which the problems are deep, rich, and enjoy many connections to other areas of mathematics and the sciences.

For instance, consider the following seemingly innocuous question: Is the set $\{ 1,1,-2/3,-2/3,-2/3 \}$ the spectrum of a 5-by-5 entrywise nonnegative matrix? Although not apparent at fi rst glance, the answer is `no’.

This question is part of the longstanding nonnegative inverse eigenvalue problem (NIEP) — one of the premier unsolved problems in matrix analysis — which can be stated more formally as follows: given a subset $\sigma = \{ \lambda_1,\dots,\lambda_n \}$ of complex numbers, find necessary and sufficient conditions such that $\latex \sigma$ is the spectrum of an entrywise nonnegative matrix. The NIEP has been open since 1949 and has attracted the attention of numerous researchers from diverse research areas, yet it remains open for all subsets $n \geq 5$.

Introduced in [1], the study of Perron similarities provides a geometric approach to investigating the NIEP (and several of its variants). After an introduction to the theory of nonnegative matrices, polyhedra, and Perron Spectratopes, students will continue to investigate the relationship between Perron similarities and solutions of the NIEP.

Specfic topics include (but are not limited to):

(i) Studying orthogonal Perron similarities in relation to the symmetric NIEP.
(ii) Investigating permutative matrices [2] and realizable spectra.
(iii) Let $p(z)$ be a polynomial whose root are realizable. C. R. Johnson posed the following question: are the roots of $p'(z)$ realizable?
(iv) establish the concavity and differentiability of the so-called Karpelevic arcs of the celebrated Karpelevic theorem.

Dr. Paparella mentored two research groups at the 2014 WM REU, and supervised summer research projects for one group in the summer of 2015 and two groups in the summer of 2016. These efforts have produced one peer-reviewed publication in the journal Linear and Multilinear Algebra and one publication currently under-review in the Minnesota Journal of Undergraduate Mathematics.

References

1. Charles R. Johnson, Pietro Paparella, Perron spectratopes and the real nonnegative inverse eigenvalue problem, Linear Algebra and its Applications, Volume 493, 15 March 2016, Pages 281-300, ISSN 0024-3795, http://dx.doi.org/10.1016/j.laa.2015.11.033.

2. Paparella, Pietro. (2016), “Realizing Suleimanova-type Spectra via Permutative Matrices”, Electronic Journal of Linear Algebra, Volume 31, pp. 306-312. DOI: http://dx.doi.org/10.13001/1081-3810.3101.