Screening In A 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 the signpattern class of A such that the characteristic polynomial of B is r(x).If replacing any nonzeroentry of A by zero destroys this property, then A is a minimal spectrally arbitrary signpattern.In this paper,we give a Nilpotent-Jacobian screening method for seeking spectrallyarbitrary sign patterns,and give examples which are spectrally arbitrary sign patterns withthe tools of Nilpotent-Jacobian screening method.In chapter 1, we introduce the history of development on the sign pattern matrices,some method in our paper had used, and our research 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 the Nilpotent-Jacobian screening method for seeking spectrallyarbitrary sign patterns,and give examples which are spectrally arbitrary sign patterns withthe tools of Nilpotent-Jacobian screening method,and we discuss that ten sign patterns areminimally spectrally arbitrary sign patterns, and every super-pattern of it is a spectrallyarbitrary sign pattern.