On universal realizability in the left half-plane

• Alfaro, Jaime H. ^[1] ; Soto, Ricardo L. ^[1]
  1. [1] Universidad Católica del Norte

     Universidad Católica del Norte

     Antofagasta, Chile

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 5, 2023, págs. 1373-1390
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-6058
• Enlaces
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• Resumen

  • A list Λ = {λ1, λ2,..., λn} of complex numbers is said to be realizable if it is the spectrum of a nonnegative matrix. Λ is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ. In this paper, using companion matrices and applying a procedure by Šmigoc, we provide sufficient conditions for the universal realizability of left half-plane spectra, that is, spectra Λ = {λ1,...,λn} with λ1 > 0, Re λi ≤ 0, i = 2, . . . , n. It is also shown how the effect of adding a negative real number to a not UR left half-plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized.

• Referencias bibliográficas
  • A. Borobia, J. Moro and R. Soto, “Negativity compensation in the nonnegative inverse eigenvalue problem”, Linear Algebra and its Applications,...
  • M. Collao, M. Salas and R. L. Soto, “Spectra universally realizable by doubly stochastic matrices”, Special Matrices, vol. 6, pp. 301-309,...
  • R. C. Diaz and R. L. Soto, “Nonnegative inverse elementary divisors problem in the left half-plane”, Linear Multilinear Algebra, vol. 64,...
  • C. R. Johnson, “Row stochastic matrices similar to doubly stochastic matrices”, Linear Multilinear Algebra, vol. 10, pp. 113-130, 1981. https://doi.org/10.1080/03081088108817402
  • C. R. Johnson, A. I. Julio and R. L. Soto, “Nonnegative realizability with Jordan structure”, Linear Algebra and its Applications, vol. 587,...
  • C. R. Johnson, A. I. Julio and R. L. Soto, “Indices of diagonalizable and universal realizability of spectra”, pre-print.
  • T. J. Laffey and H. Šmigoc, “Nonnegative realization of spectra having negative real parts”, Linear Algebra and its Applications, vol. 416,...
  • H. Minc, “Inverse elementary divisor problem for nonnegative matrices”, Proceedings of the American Mathematical Society, vol. 83, pp. 665-669,...
  • H. Minc, “Inverse elementary divisor problem for doubly stochastic matrices”, Linear Multilinear Algebra, vol. 11, pp. 121-131, 1982. https://doi.org/10.1080/03081088208817437
  • H. Šmigoc, “The inverse eigenvalue problem for nonnegative matrices”, Linear Algebra and its Applications, vol. 393, pp. 365-374, 2004. https://doi.org/10.1016/j.laa.2004.03.036
  • R. L. Soto, R. C. Díaz, H. Nina, and M. Salas, “Nonnegative matrices with prescribed spectrum and elementary divisors”, Linear Algebra and...