Realizability by symmetric nonnegative matrices

• Soto Montero, Ricardo Lorenzo ^[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. 24, Nº. 1, 2005, págs. 65-78
• Idioma: inglés
• DOI: 10.4067/S0716-09172005000100006
• Enlaces
  • Texto completo
• Resumen

  • Let Λ = {λ1, λ2,...,λn} be a set of complex numbers. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions in order that Λ may be the spectrum of an entrywise nonnegative n × n matrix. If there exists a nonnegative matrix A with spectrum Λ we say that Λ is realized by A. If the matrix A must be symmetric we have the symmetric nonnegative inverse eigenvalue problem (SNIEP). This paper presents a simple realizability criterion by symmetric nonnegative matrices. The proof is constructive in the sense that one can explicitly construct symmetric nonnegative matrices realizing Λ.

• Referencias bibliográficas
  • Citas [1] A. Borobia, On the Nonnegative Eigenvalue Problem, Linear Algebra Appl. 223-224, pp. 131-140, (1995).
  • [2] A. Borobia, J. Moro, R. Soto, Negativity compensation in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 393, pp. 73-89,...
  • [3] A. Brauer, Limits for the characteristic roots of a matrix. IV: Aplications to stochastic matrices, Duke Math. J., 19, pp. 75-91, (1952).
  • [4] M. Fiedler, Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl. 9, pp. 119-142, (1974).
  • [5] C. R. Johnson, T. J. Laffey, R. Loewy, The real and the symmetric nonnegative inverse eigenvalue problems are different, Proc. AMS 124,...
  • [6] R. Kellogg, Matrices similar to a positive or essentially positive matrix, Linear Algebra Appl. 4, pp. 191-204, (1971).
  • [7] T. J. Laffey, E. Meehan, A characterization of trace zero nonnegative 5x5 matrices, Linear Algebra Appl. pp. 302-303, pp. 295-302, (1999).
  • [8] R. Loewy, D. London, A note on an inverse problem for nonnegative matrices, Linear and Multilinear Algebra 6, pp. 83-90, (1978).
  • [9] H. Perfect, Methods of constructing certain stochastic matrices, Duke Math. J. 20, pp. 395-404, (1953).
  • [10] N. Radwan, An inverse eigenvalue problem for symmetric and normal matrices, Linear Algebra Appl. 248, pp. 101-109, (1996).
  • [11] R. Reams, An inequality for nonnegative matrices and the inverse eigenvalue problem, Linear and Multilinear Algebra 41, pp. 367-375,...
  • [12] F. Salzmann, A note on eigenvalues of nonnegative matrices, Linear Algebra Appl. 5., pp. 329-338, (1972).
  • [13] R. Soto, Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra Appl. 369, pp. 169-184, (2003).
  • [14] R. Soto, A. Borobia, J. Moro, On the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem, Linear...
  • [15] G. Soules, Constructing symmetric nonnegative matrices, Linear and Multilinear Algebra 13, pp. 241-251, (1983).
  • [16] H. R. Suleimanova, Stochastic matrices with real characteristic values, Dokl. Akad. Nauk SSSR 66, pp. 343-345, (1949).
  • [17] G. Wuwen, An inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl. 249, pp. 67-78, (1996).