Allow problems concerning spectral properties of patterns

Abstract

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.

Techniques for identifying inertially arbitrary patterns

“…The next result, which was first proved in [3], follows from Corollary 3.6 with a + = a − = 0 and gives a technique that may be used to show a pattern is inertially arbitrary.…”

“…Therefore, by induction B = O and by Theorem 4.9, A and all of its superpatterns are inertially arbitrary. Example 4.13 addresses a remark made in [3]. In particular, this example shows that there exists an inertially arbitrary sign pattern that does not allow a positive and a negative principal minor of every order, correcting a statement made in [8].…”

Counterexamples on spectra of sign patterns

“…The study of spectra of matrix patterns deserved a significant amount of attention in recent publications. The conjecture mentioned in the abstract appeared in one of the foundational papers on this topic ( [10]), and many subsequent works proved it in different special cases ( [3,7,12,14,15]). One of the known sufficient conditions for superpatterns to be spectrally arbitrary is the Nilpotent Jacobian condition ( [2,10]), which allowed to solve several intriguing problems on this topic ( [4,11,19]).…”

Spectrally arbitrary pattern extensions

“…signed triangle extension of W n ) for all n ≥ 6. Further, W * 5 allows refined inertia (0, 0, 1, 4) as noted in [6] (and likewise W 6 allows refined inertia (0, 0, 2, 4) as noted in [5]). By recursive application of Theorem 4.2, we can observe that every superpattern of W n and W * n−1 is inertially arbitrary for all n ≥ 6.…”