On The Spectral Properties Of Matrix Patterns

As an important part of combinatorial matrix theory, sign patterns and zero-nonzeropatterns study mainly on the combinatorial properties that base solely on the signs and thezero-nonzero position of entries in a matrix in its qualitative class, regardless of the mag-nitudes of its nonzero entries. The researches on sign patterns and zero-nonzero patternshave been much applied in control system theory, ecology, chemistry, theoretic computerscience, economics and social science, etc. In this dissertation, some spectral propertiesof sign patterns and zero-nonzero patterns have been investigated. It includes the eventualpositivity, power-positivity, nilpotence, spectrally and inertially arbitrary property and therefined inertially arbitrary property. This thesis is divided into three parts, including fivechapters.1. We first come up with two concepts to study potentially eventually positive signpatterns. One is the minimal potentially eventually positive sign pattern, and the other isthe minimal potentially power-positive sign pattern. Some sufficient or necessary condi-tions for a sign pattern to be a minimal potentially eventually positive or power-positivesign pattern are established. Based on these results, the classification of all minimal po-tentially eventually positive sign patterns of order?3 has been handled completely. Anote has been given on the 4×4 potentially eventually positive sign patterns. Moreover,we have identified the potentially eventually positive star and double star sign patternsas the superpatterns of some minimal potentially eventually positive star and double starsign patterns, respectively.2. We investigate spectral properties of some 4×4 irreducible tridiagonal sign pat-terns. Some necessary conditions for a 4×4 irreducible tridiagonal sign pattern to bea potentially nilpotent sign pattern are established first. Then we show that a 4×4 ir-reducible tridiagonal sign pattern with two nonzero diagonal entries or three successivenonzero diagonal entries is potentially nilpotent if and only if it is a spectrally arbitrarysign pattern. It partially answers an open question raised in the recent paper. Finally,we introduce a new class of inertially arbitrary zero-nonzero patterns with less than 2n(n?5) nonzero entries and a class of inertially arbitrary sign patterns with less than 2n(n?6) nonzero entries. 3. Based on a splitting of matrix inertias, we come up with minimal critical setsof refined inertias to investigate the critical sets of inertias for irreducible zero-nonzeropatterns and sign patterns. All minimal critical sets of inertias for irreducible zero-nonzeropatterns of order?3 have been identified by a careful discussion on the minimal criticalsets of refined inertias. Also identified is the cardinality number of the minimal criticalsets of inertias for 4×4 irreducible zero-nonzero patterns.