Allow problems concerning spectral properties of patterns
Michael Cavers, Shaun Fallat · Electronic Journal of Linear Algebra · 2012
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.