Borobia Vizmanos, AlbertoCanogar Mckenzie, Roberto2025-12-052025-12-052021-03-05Borobia, A., & Canogar, R. (2021). Nonsparse companion hessenberg matrices. Electronic Journal of Linear Algebra, 37, 193-210. https://doi.org/10.13001/ELA.2021.54051537-9582https://doi.org/10.13001/ela.2021.5405https://hdl.handle.net/20.500.14468/31037The registered version of this article, first published in Electronic Journal of Linear Algebra, is available online at the publisher's website: International Linear Algebra Society (ILAS), https://doi.org/10.13001/ela.2021.5405La versión registrada de este artículo, publicado por primera vez en Electronic Journal of Linear Algebra, está disponible en línea en el sitio web del editor: International Linear Algebra Society (ILAS), https://doi.org/10.13001/ela.2021.5405In recent years, there has been a growing interest in companion matrices. Sparse companion matrices are well known: every sparse companion matrix is equivalent to a Hessenberg matrix of a particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223247, 2019] started the systematic study of nonsparse companion matrices. They proved that every nonsparse companion matrix is nonderogatory, although not necessarily equivalent to a Hessenberg matrix. In this paper, the nonsparse companion matrices which are unit Hessenberg are described. In a companion matrix, the variables are the coordinates of the characteristic polynomial with respect to the monomial basis. A PB-companion matrix is a generalization, in the sense that the variables are the coordinates of the characteristic polynomial with respect to a general polynomial basis. The literature provides examples with Newton basis, Chebyshev basis, and other general orthogonal bases. Here, the PB-companion matrices which are unit Hessenberg are also described.eninfo:eu-repo/semantics/openAccess12 MatemáticasartículoCompanion matrixCharacteristic polynomialHessenberg matrixPolynomial basisNilpotent matrix