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# On the characterization of totally nonpositive matrices

Cantó Colomina, R.; Pelaez, MJ.; Urbano Salvador, AM. (2016). On the characterization of totally nonpositive matrices. SeMA Journal. 73(4):347-368. doi:10.1007/s40324-016-0073-1.

Por favor, use este identificador para citar o enlazar este ítem: http://hdl.handle.net/10251/83249

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 Título: On the characterization of totally nonpositive matrices Autor: Entidad UPV: Universitat Politècnica de València. Escuela Técnica Superior de Ingeniería Agronómica y del Medio Natural - Escola Tècnica Superior d'Enginyeria Agronòmica i del Medi Natural Universitat Politècnica de València. Escuela Politécnica Superior de Alcoy - Escola Politècnica Superior d'Alcoi Fecha difusión: 2016-12 Resumen: A nonpositive real matrix $A= (a_{ij})_{1 \leq i, j \leq n}$ is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal ...[+] A nonpositive real matrix $A= (a_{ij})_{1 \leq i, j \leq n}$ is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal factorization of a nonsingular t.n.p. matrix $A$ is computed and it is stored in an matrix represented by $\mathcal{BD}_{(t.n.p.)}(A)$ when $a_{11}< 0$ (or $\mathcal{BD}_{(zero)}(A)$ when $a_{11}= 0$). As a converse result, an efficient algorithm to know if an matrix $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(zero)}(A)$) is the bidiagonal factorization of a t.n.p. matrix with $a_{11}<0$ ($a_{11}= 0$) is given. Similar results are obtained for t.n. matrices using the matrix $\mathcal{BD}_{(t.n.)}(A)$, and these characterizations are extended to rectangular t.n.p. (t.n.) matrices. Finally, the bidiagonal factorization of the inverse of a nonsingular t.n.p. (t.n.) matrix $A$ is directly obtained from $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(t.n.)}(A)$). [-] Palabras clave: Derechos de uso: Embargado Fuente: SeMA Journal. (issn: 2254-3902 ) (eissn: 2281-7875 ) DOI: 10.1007/s40324-016-0073-1 Editorial: Springer Versión del editor: http://dx.doi.org/10.1007/s40324-016-0073-1 Descripción: “The final publication is available at Springer via http://dx.doi.org/10.1007/s40324-016-0073-1” Patrocinador: Spanish DGI grant/MTM2013-43678-P Chile/FONDECYT 1100029 Tipo: Artículo