Upper Hessenberg and Toeplitz Bohemians
Authors
Chan, Eunice Y.S.; Corless, Robert M.; González Vega, Laureano; Sendra Pons, Juan Rafael; Sendra Pons, Juana; [et al.]Identifiers
Permanent link (URI): http://hdl.handle.net/10017/55314DOI: 10.1016/j.laa.2020.03.037
ISSN: 0024-3795
Publisher
Elsevier
Date
2020-09-15Funders
Agencia Estatal de Investigación
Bibliographic citation
Chan, E.Y.S., Corless, R.M., Gonzalez-Vega, L., Sendra, J.R., Sendra, J. & Thornton, S.E. 2020, “Upper Hessenberg and Toeplitz Bohemians”, Linear Algebra and its Applications, vol. 601, pp. 72-100.
Keywords
Upper Hessenberg
Toeplitz
Characteristic polynomial
Bohemians
Maximal characteristic height
Normal matrices
Stable matrices
Description / Notes
We also acknowledge the support of the Ontario Graduate Institution, The National Science & Engineering Research Council of Canada, the University of Alcala, the Rotman Institute of Philosophy, the Ontario Research Centre of Computer Algebra, and Western University. Part of this work was developed while R. M. Corless was visiting the University of Alcala, in the frame of the project Giner de los Rios. J.R. Sendra is member of the Research Group ASYNACS (Ref. CT-CE2019/683).
Project
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-88796-P/ES/COMPUTACION SIMBOLICA: NUEVOS RETOS EN ALGEBRA Y GEOMETRIA Y SUS APLICACIONES/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/acceptedVersion
Publisher's version
https://doi.org/10.1016/j.laa.2020.03.037Rights
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
© 2020 Elsevier
Access rights
info:eu-repo/semantics/openAccess
Abstract
A set of matrices with entries from a fixed finite population P is called “Bohemian”. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population {−1,0,+1} and sometimes other populations, for instance {0,1,i,−1,−i}. More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families.
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