Sparse inertially arbitrary patterns
| Autor: | Loretta Vanderspek, Michael S. Cavers, Kevin N. Vander Meulen |
|---|---|
| Rok vydání: | 2009 |
| Předmět: |
Inertia
media_common.quotation_subject 0211 other engineering and technologies 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Combinatorics Matrix (mathematics) Spectrum Nonzero pattern Discrete Mathematics and Combinatorics Potentially nilpotent 0101 mathematics Mathematics media_common Characteristic polynomial Numerical Analysis Algebra and Number Theory Spectrum (functional analysis) 021107 urban & regional planning Graph theory Inverse problem Algebra Geometry and Topology Nilpotent group Sign pattern Sign (mathematics) |
| Zdroj: | Linear Algebra and its Applications. 431(11):2024-2034 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2009.06.040 |
| Popis: | An n-by-n sign pattern A is a matrix with entries in {+,-,0}. An n-by-n nonzero pattern A is a matrix with entries in {∗,0} where ∗ represents a nonzero entry. A pattern A is inertially arbitrary if for every set of nonnegative integers n1,n2,n3 with n1+n2+n3=n there is a real matrix with pattern A having inertia (n1,n2,n3). We explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials. This information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary. Britz et al. [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] conjectured that irreducible spectrally arbitrary patterns must have at least 2n nonzero entries; we demonstrate that irreducible inertially arbitrary patterns can have less than 2n nonzero entries. |
| Databáze: | OpenAIRE |
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