Spectrally arbitrary star sign patterns !
نویسندگان
چکیده
An n × n sign pattern Sn is spectrally arbitrary if, for any given real monic polynomial g(x) of degree n, there is a real matrix having sign pattern Sn and characteristic polynomial g(x). All n × n star sign patterns that are spectrally arbitrary, and all minimal such patterns, are characterized. This subsequently leads to an explicit characterization of all n × n star sign patterns that are potentially nilpotent. It is shown that any super-pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary. © 2004 Elsevier Inc. All rights reserved.
منابع مشابه
Spectrally and inertially arbitrary sign patterns
We introduce some n-by-n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia. Consequently, we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary. We demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3. We illustrate that in general, ...
متن کاملMinimal Spectrally Arbitrary Sign Patterns
An n × n sign pattern A is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of A with that spectrum. If replacing any nonzero entry (or entries) of A by zero destroys this property, then A is a minimal spectrally arbitrary sign pattern. For n ≥ 3, several families of n × n spectrally arbitrary sign patterns are presented, and their minimal spectrally a...
متن کاملNilpotent matrices and spectrally arbitrary sign patterns
It is shown that all potentially nilpotent full sign patterns are spectrally arbitrary. A related result for sign patterns all of whose zeros lie on the main diagonal is also given.
متن کاملInertially arbitrary nonzero patterns of order 4
Inertially arbitrary nonzero patterns of order at most 4 are characterized. Some of these patterns are demonstrated to be inertially arbitrary but not spectrally arbitrary. The order 4 sign patterns which are inertially arbitrary and have a nonzero pattern that is not spectrally arbitrary are also described. There exists an irreducible nonzero pattern which is inertially arbitrary but has no si...
متن کاملInertially arbitrary tree sign patterns of order 4
An n × n sign pattern matrix A is an inertially arbitrary pattern if for every nonnegative triple (n1, n2, n3) with n1 + n2 + n3 = n, there is a real matrix in the sign pattern class of A having inertia (n1, n2, n3). An n× n sign pattern matrix A is a spectrally arbitrary pattern if for any given real monic polynomial r(x) of degree n, there is a real matrix in the sign pattern class of A with ...
متن کامل