Artículos científicos
Una condición recursiva para el problema inverso del autovalor para matrices simétricas no negativas
Publicado 2017-08-09
Palabras clave
- Problemas inversos,
- autovalores,
- matrices ortogonales,
- matrices simétricas
Cómo citar
Valero, E. R., Mallea-Zepeda, E., & Lenes, E. (2017). Una condición recursiva para el problema inverso del autovalor para matrices simétricas no negativas. Revista Integración, Temas De matemáticas, 35(1), 37–50. https://doi.org/10.18273/revint.v35n1-2017003
Resumen
En este artículo presentamos una condición sufiente 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 Λ= {λ1, …, λn, λn+1} es realizable por una matriz simétrica no negativa, si la lista μ = {μ1, ..., μn} asociada a Λ es realizable. Este resultado es fácil de programar y mejora algunos criterios existentes.
MSC2010: 15A29, 15A18, 15B10, 15A57.
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Referencias
1. Ellard, Richard, and Helena Šmigoc. "Connecting sufficient conditions for the symmetric nonnegative inverse eigenvalue problem." Linear Algebra and its Applications 498 (2016): 521-552.
2. Fiedler, Miroslav. "Eigenvalues of nonnegative symmetric matrices." Linear Algebra and its Applications 9 (1974): 119-142.
3. Guo, Wuwen. "An inverse eigenvalue problem for nonnegative matrices." Linear Algebra and its Applications 249.1-3 (1996): 67-78.
4. Horn A. and Johnson C.R., Matrix analysis, Cambridge university press, Cambridge, 1990. Corrected reprint of the 1985 original
5. Johnson, C.R., Laffey T.J. and Loewy R., "The real and the symmetric nonnegative inverse eigenvalue problems are different.", Proc. Amer. Math. Soc. 124 (1996), No. 12, 3647-3651.
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.
7. Loewy R. and London D., "A note on an inverse problem for nonnegative matrices", Linear and Multilinear Algebra 6 (1978), No. 1, 83-90.
8. Loewy R. and McDonald J.J., "The symmetric nonnegative inverse eigenvalue problem for 5×5 matrices", Linear algebra appl. 393 (2004), 275-298.
9. Meehan M.E., “Some results on the matrix spectra”, Thesis (Ph.D.), National University of Ireland, Dublin, 1998.
10. McDonald J.J. and Neumann M., "The Soules approach to the inverse eigenvalue problem for nonnegative symmetric matrices of order n ≤ 5", in Contemp. Math. 259, Amer. Math. Soc. (2000), 387-408.
11. Radwan N., "An inverse eigenvalue problem for symmetric and normal matrices.", Linear algebra appl. 248 (1996), 101-109.
12. Soto R.L., "A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem", Numer. Linear Algebra Appl 20 (2013), 336-348.
13. Soto R.L., "Realizability criterion for the symmetric nonnegative inverse eigenvalue problem.", Linear Algebra appl. 416 (2006), No. 2-3, 783-794.
14. Soto, Ricardo L., and Elvis Valero. "On Symmetric Nonnegative Matrices with Prescribed Spectrum" International Mathematical Forum 9 (2014), No. 24, 1161-1176.
15. Soules G.W., "Constructing symmetric nonnegative matrices.", Linear Multilinear Algebra 13 (1983), No. 2, 241-251.
16. Spector O., "A characterization of trace zero symmetric nonnegative 5× 5 matrices" Linear Algebra Appl 434 (2011), No. 4, 1000-1017.
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. EBL digraphs." Linear Algebra Appl. 426 (2007), No. 2-3, 729-773.
2. Fiedler, Miroslav. "Eigenvalues of nonnegative symmetric matrices." Linear Algebra and its Applications 9 (1974): 119-142.
3. Guo, Wuwen. "An inverse eigenvalue problem for nonnegative matrices." Linear Algebra and its Applications 249.1-3 (1996): 67-78.
4. Horn A. and Johnson C.R., Matrix analysis, Cambridge university press, Cambridge, 1990. Corrected reprint of the 1985 original
5. Johnson, C.R., Laffey T.J. and Loewy R., "The real and the symmetric nonnegative inverse eigenvalue problems are different.", Proc. Amer. Math. Soc. 124 (1996), No. 12, 3647-3651.
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.
7. Loewy R. and London D., "A note on an inverse problem for nonnegative matrices", Linear and Multilinear Algebra 6 (1978), No. 1, 83-90.
8. Loewy R. and McDonald J.J., "The symmetric nonnegative inverse eigenvalue problem for 5×5 matrices", Linear algebra appl. 393 (2004), 275-298.
9. Meehan M.E., “Some results on the matrix spectra”, Thesis (Ph.D.), National University of Ireland, Dublin, 1998.
10. McDonald J.J. and Neumann M., "The Soules approach to the inverse eigenvalue problem for nonnegative symmetric matrices of order n ≤ 5", in Contemp. Math. 259, Amer. Math. Soc. (2000), 387-408.
11. Radwan N., "An inverse eigenvalue problem for symmetric and normal matrices.", Linear algebra appl. 248 (1996), 101-109.
12. Soto R.L., "A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem", Numer. Linear Algebra Appl 20 (2013), 336-348.
13. Soto R.L., "Realizability criterion for the symmetric nonnegative inverse eigenvalue problem.", Linear Algebra appl. 416 (2006), No. 2-3, 783-794.
14. Soto, Ricardo L., and Elvis Valero. "On Symmetric Nonnegative Matrices with Prescribed Spectrum" International Mathematical Forum 9 (2014), No. 24, 1161-1176.
15. Soules G.W., "Constructing symmetric nonnegative matrices.", Linear Multilinear Algebra 13 (1983), No. 2, 241-251.
16. Spector O., "A characterization of trace zero symmetric nonnegative 5× 5 matrices" Linear Algebra Appl 434 (2011), No. 4, 1000-1017.
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. EBL digraphs." Linear Algebra Appl. 426 (2007), No. 2-3, 729-773.