Sign Pattern Matrices

Abstract

The study of sign pattern matrices is an important part of combinatorial matrix theory. It has a rich theory in its own right and, in addition, some results are useful in applications to dynamical systems where sign patterns arise naturally, for example in predator-prey populations, economics, chemistry, and sociology. The aim of this chapter is to describe some spectral properties of matrices with a given sign pattern from the perspective of combinatorial matrix theory, showing some results, techniques, applications, and open problems. Topics are chosen from the literature on sign patterns; some of these and other related topics can be found in the book Brualdi–Shader [7], and in the chapter Hall–Li [34] from the Handbook of Linear Algebra edited by L. Hogben, and references therein. Readers are encouraged to consult these other references for topics beyond those in this chapter, which is a somewhat personal overview of some results on sign pattern matrices.