Services on Demand
Journal
Article
Indicators
- Cited by SciELO
- Access statistics
Related links
- Cited by Google
- Similars in SciELO
- Similars in Google
Share
Revista Integración
Print version ISSN 0120-419X
Integración - UIS vol.35 no.1 Bucaramanga Jan./June 2017
https://doi.org/10.18273/revint.v35n1-2017003
Original article
A recursive condition for the symmetric nonnegative inverse eigenvalue problema
Una condición recursiva para el problema inverso del autovalor para matrices simétricas no negativas
1Universidad de Tarapacá, Departamento de Matemática, Arica, Chile.
2Universidad del Sinú. Elías Bechara Zainum, Departamento de Matemática, Cartagena, Colombia
In this paper we present a sufficient condition and a necessary condition for Symmetric Nonnegative Inverse Eigenvalue Problem. This condition is independent of the existing realizability criteria. This Criterion is recursive, that is, it determines whether a list A = {Al, ... , An, An+d is realizable by a nonnegative symmetric matrix, if the list M = {MI, ... , Mn} associated to A is realizable. This result is easy to program and improves sorne existing criteria.
Keywords: Inverse problems; eigenvalues; orthogonal matrices; symmetric matrix
En este artículo presentamos una condición suficiente y una condición necesaria para el Problema Inverso de Autovalores para Matrices Simétricas no Negativas. Esta condición es independiente de los criterios de realizabilidad existentes. Este criterio es recursivo, es decir determina si una lista A = {Al, ... , An, An+d es realizable por una matriz simétrica no negativa, si la lista M = {MI, ... , Mn} asociada a A es realizable. Este resultado es fácil de programar y mejora algunos criterios existentes.
Palabras clave: Problemas inversos; autovalores; matrices ortogonales; matrices simétricas
References
1. Ellard R. and Smigoc H., "Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalues Problem", Linear Algebra Appl. 498 (2016), 521-552. [ Links ]
2. Fiedler M., "Eigenvalues of Nonnegative Symmetric Matrices", Linear Algebra Appl. 9 (1974),119-142. [ Links ]
3. Guo W., "An Inverse Eigenvalues Problem for Nonnegative Matrices", Linear Algebra Appl. 249 (1996), No. 1-3, 67- 78. [ Links ]
4. Horn A. and Johnson C.R. , Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. [ Links ]
5. Johnson C.R., Laffey T.J. and Loewy R., "The real and the symmetric nonnegative inverse eigenvalue problems are different", Proc. Amer. Math. Soco 124 (1996), No. 12, 3647- 3651. [ Links ]
6. Johnson C.R., Marijuán C. and Pisonero M. , "Ruling out certain 5-spectra for the symmetric nonnegative inverse eigenvalue problem", Linear Algebra Appl. 512 (2017) , 129-135. [ Links ]
7. Loewy R. and London D. , "A note on an inverse problem for nonnegative matrices", Linear Multilinear Algebra 6 (1978) , No. 1, 83- 90. [ Links ]
8. Loewy R. and McDonald J .J. , "The symmetric nonnegative inverse eigenvalue problem for 5 x 5 matrices", Linear Algebm Appl. 393 (2004) , 275- 298. [ Links ]
9. Meehan M.E. , "Some results on the matrix spectra", Thesis (Ph.D.) , National University of Ireland, Dublin, 1998. [ Links ]
10. McDonald J.J. and Neumann M. , "The Soules approach to the inverse eigenvalues problem for nonnegative symmetric matrices of order n ≤ 5 ", in Contemp. Math. 259, Amer. Math. Soc. (2000), 387- 407. [ Links ]
11. Radwan N., "An Inverse eigenvalue problem for symmetric and normal matrices", Linear Algebm Appl. 248 (1996) , 101- 109. [ Links ]
12. Soto R.L., "A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem", Numer. Linear Algebm Appl. 20 (2013), No. 2, 336-348. [ Links ]
13. Soto R.L., "Realizability criterion for the symmetric nonnegative inverse eigenvalue problem", Linear Algebm Appl. 416 (2006), No. 2-3, 783- 794. [ Links ]
14. Soto R.L. and Valero E. , "On Symmetric Nonnegative matrices with Prescribed Spectrum", International Mathematical Forum 9 (2014), No. 24, 1161- 1176. [ Links ]
15. Soules G.,"Constructing symmetric nonnegative matrices", Linear Multilinear Algebm 13 (1983), No. 3, 241- 251. [ Links ]
16. Spector O., "A characterization of trace zero symmetric nonnegative 5 x 5 matrices", Linear Algebm Appl. 434 (2011) , No. 4,1000-1017. [ Links ]
17. Torre-Mayo J ., Abril-Raymundo M.R. , Alarcia-Estévez E., Marijuán C., and Pisonero M., "The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. ELE digraphs", Linear Algebm Appl. 426 (2007) , No. 2-3, 729- 773. [ Links ]
Received: March 2017; Accepted: May 2017