## versión impresa ISSN 0716-0917

### Proyecciones (Antofagasta) v.24 n.1 Antofagasta mayo 2005

#### http://dx.doi.org/10.4067/S0716-09172005000100006

 Proyecciones Vol. 24, No 1, pp. 65-78, May 2005. Universidad Católica del Norte Antofagasta - Chile REALIZABILITY BY SYMMETRIC NONNEGATIVE MATRICES* RICARDO L. SOTO Universidad Católica del Norte, Chile. ABSTRACT Let Λ= {λ1, λ2, . . . , λn} be a set of complex numbers. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and su.cient conditions in order that Λmay 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. Key words: symmetric nonnegative inverse eigenvalue problem. AMS classification: 15A18, 15A51. References [1] A. Borobia, On the Nonnegative Eigenvalue Problem, Linear Algebra Appl. 223-224, pp. 131-140, (1995).        [ Links ][2] A. Borobia, J. Moro, R. Soto, Negativity compensation in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 393, pp. 73-89,(2004).        [ Links ][3] A. Brauer, Limits for the characteristic roots of a matrix. IV: Aplications to stochastic matrices, Duke Math. J., 19, pp. 75-91, (1952).        [ Links ][4] M. Fiedler, Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl. 9, pp. 119-142, (1974).        [ Links ][5] C. R. Johnson, T. J. La.ey, R. Loewy, The real and the symmetric nonnegative inverse eigenvalue problems are di.erent, Proc. AMS 124, pp. 3647-3651, (1996).        [ Links ][6] R. Kellogg, Matrices similar to a positive or essentially positive matrix, Linear Algebra Appl. 4, pp. 191-204, (1971).        [ Links ][7] T. J. La.ey, E. Meehan, A characterization of trace zero nonnegative 5x5 matrices, Linear Algebra Appl. pp. 302-303, pp. 295-302, (1999).        [ Links ][8] R. Loewy, D. London, A note on an inverse problem for nonnegative matrices, Linear and Multilinear Algebra 6, pp. 83-90, (1978).        [ Links ][9] H. Perfect, Methods of constructing certain stochastic matrices, Duke Math. J. 20, pp. 395-404, (1953).        [ Links ][10] N. Radwan, An inverse eigenvalue problem for symmetric and normal matrices, Linear Algebra Appl. 248, pp. 101-109, (1996).        [ Links ][11] R. Reams, An inequality for nonnegative matrices and the inverse eigenvalue problem, Linear and Multilinear Algebra 41, pp. 367-375, (1996).        [ Links ][12] F. Salzmann, A note on eigenvalues of nonnegative matrices, Linear Algebra Appl. 5., pp. 329-338, (1972).         [ Links ][13] R. Soto, Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra Appl. 369, pp. 169-184, (2003).        [ Links ][14] R. Soto, A. Borobia, J. Moro, On the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem, Linear Algebra Appl. 396, pp. 223-241, (2005).        [ Links ][15] G. Soules, Constructing symmetric nonnegative matrices, Linear and Multilinear Algebra 13, pp. 241-251, (1983).        [ Links ][16] H. R. Suleimanova, Stochastic matrices with real characteristic values, Dokl. Akad. Nauk SSSR 66, pp. 343-345, (1949).        [ Links ][17] G. Wuwen, An inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl. 249, pp. 67-78, (1996).        [ Links ]Received : March 2005, Accepted : May 2005. *Supported by Fondecyt 1050026, Chile. Ricardo L. Soto Departamento de Matemáticas Universidad Católica del Norte Casilla 1280 Antofagasta Chile e-mail : rsoto@ucn.cl

Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons