Allow problems concerning spectral properties of patterns

Abstract

Let S � {0,+, ,+0, 0,�,#} be a set of symbols, where + (resp., , +0 and 0) denotes a positive (resp., negative, nonnegative and nonpositive) real number, and � (resp., #) denotes a nonzero (resp., arbitrary) real number. An S-pattern is a matrix with entries in S. In particular, a {0,+,} -pattern is a sign pattern and a {0,�}-pattern is a zero-nonzero pattern. This paper extends the following problems concerning spectral properties of sign patterns and zero-nonzero patterns to S-patterns: spectrally arbitrary patterns; inertially arbitrary patterns; refined inertially arbitrary patterns; potentially nilpotent patterns; potentially stable patterns; and potentially purely imaginary patterns. Relationships between these classes of S-patterns are given and techniques that appear in the literature are extended. Some interesting examples and properties of patterns when # belongs to the symbol set are highlighted. For example, it is shown that there is a {0,+,#}-pattern of order n that is spectrally arbitrary with exactly 2n 1 nonzero entries. Finally, a modified version of the nilpotent-Jacobian method is presented that can be used to show a pattern is inertially arbitrary. 1. Introduction: Definitions, background and motivation. 1.1. Definitions. Throughout this paper, we assume all matrices are square and use the notation S to denote the set of symbols S = f0;+;−;+0;−0;�;#g, where + (resp., −) represents a positive (resp., negative) real number, +0 (resp., −0) represents a nonnegative (resp., nonpositive) real number, and � (resp., #) represents a nonzero (resp., arbitrary) real number. For a symbol set SS, an S-pattern is a matrix with entries in S. In particular, a f0;+;−g-pattern is a sign pattern, a f0;�g-pattern is a zero-nonzero pattern, a f+;−g-pattern is a full sign pattern, a f0;+g-pattern is a nonnegative sign pattern, a f+g-pattern is a positive sign pattern, and a f0;+;−;#g- pattern is a generalized sign pattern. We use the term pattern when statements hold for all S-patterns with SS.