The Minimal Spectrally Arbitrary Pattern

A sign pattern matrix A of order n is a spectrally arbitrary pattern (SAP) if given anymonic polynomial r(x) of order n with real coe?cients, there exists a real matrix B in thesign pattern class of A such that the characteristic polynomial of B is r(x).If replacing anynonzero entry of A by zero destroys this property, then A is a minimal spectrally arbitrarysign pattern.In this paper,we give a sign patten and a class of sign patten are minimalspectrally arbitrary pattern by using the Nilpotent-Jacobian method.In chapter 1, we introduce the history of development on the sign pattern matrices, ourresearch problems and main results.In chapter 2, a sign patten which is spectrally arbitrary is investigated by using theNilpotent-Jacobian method. Furthermore,we demonstrate that it is actually minimal spec-trally arbitrary pattern,and every superpattern of it is a spectrally arbitrary sign pattern.In chapter 3, we give a class of sign pattens which are spectrally arbitrary by usingthe Nilpotent-Jacobian method,Furthermore,we demonstrate that they are actually minimalspectrally arbitrary patterns,and their superpatterns are spectrally arbitrary sign patterns.