Research and Innovation Articles
A recursive condition for the symmetric nonnegative inverse eigenvalue problem
Published 2017-08-09
Keywords
- Inverse problems,
- eigenvalues,
- orthogonal matrices,
- symmetric matrix
How to Cite
Valero, E. R., Mallea-Zepeda, E., & Lenes, E. (2017). A recursive condition for the symmetric nonnegative inverse eigenvalue problem. Revista Integración, Temas De matemáticas, 35(1), 37–50. https://doi.org/10.18273/revint.v35n1-2017003
Abstract
In this paper we present a sufficient ondition and a necessary condition for Symmetri Nonnegative Inverse Eigenvalue Problem. This condition is independent of the existing realizability criteria. This criterion is recursive, that is, it determines whether a list Λ= {λ1, …, λn, λn+1} is realizable by a nonnegative symmetric matrix, if the list μ = {μ1, ..., μn} associated to Λ is realizable. This result is easy to program and improves some existing criteria.
MSC2010: 15A29, 15A18, 15B10, 15A57.
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References
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.