On Open Problems of Nonnegative Inverse Eigenvalues Problem
Jun-Liang Wu
DOI: 10.4236/apm.2011.14025   PDF    HTML     7,125 Downloads   15,627 Views   Citations


In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.

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J. Wu, "On Open Problems of Nonnegative Inverse Eigenvalues Problem," Advances in Pure Mathematics, Vol. 1 No. 4, 2011, pp. 128-132. doi: 10.4236/apm.2011.14025.

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The authors declare no conflicts of interest.


[1] A. N. Kolmogorov, “Markov Chains with Countably Many Possible States,” Bull University, Moscow, 1937, pp. 1-16.
[2] H. Minc, “Nonnegative Matrices,” John Wiley and Sons, New York, 1988, p. 166.
[3] K. R. Suleimanova, “Stochastic Matrices with Real Eigenvalues,” Soviet Mathematics Doklady, Vol. 66, 1949, pp. 343-345.
[4] R. Loewy and D. London, “A Note on an Inverse Problem for Nonnega-tive Matrices,” Linear and Multilinear Algebra, Vol. 6, No. 1, 1978, pp. 83-90. doi:10.1080/03081087808817226
[5] M. Fiedler, “Eigen-values of Nonnegative Symmetric Matrices,” Linear Algebra Applied, Vol. 9, 1974, pp. 119-142. doi:10.1016/0024-3795(74)90031-7UUU
[6] M. T. Chu and G. H. Golub, “Inverse Eigenvalue Problems: Theory, Algorithms, and Applications,” Oxford University, Oxford, pp. 93-122.
[7] R. L. SoTo, “Reliability by Symmetric Nonnegative Matrices,” http://www.scielo.cl/pdf/proy/v24n1/art06.pdf.
[8] A. Boro-bia, “On the Nonnegative Eigenvalue Problem,” Linear Alge-bra Applied, Vol. 223-224, 1995, pp. 131-140. doi:10.1016/0024-3795(94)00343-CUUU
[9] M. Boyle and D. Handelman, “The Spectra of Nonnegative Matrices via Sym-bolic Dynamics,” Annals of Mathematics, Vol. 133, 1991, pp. 249-316. doi:10.2307/2944339
[10] P. Egleston, “Nonnegative Matrices with Prescribed Spectra,” Dissertation, Central Michigan University, 2001.
[11] C. Johnson, “Row Stochastic Matrices Similar to Doubly Stochas-tic Matrices,” Linear and Multilinear Algebra, Vol. 10, No. 2, 1981, pp. 113-130. doi:10.1080/03081088108817402
[12] C. Johnson, T. J. Laffey and R. Loewy, “The Real and the Symmetric Nonnega-tive Inverse Eigenvalue Problems are Different,” Proceedings of the American Mathematical Society, Vol. 124, 1996, PP. 3647-3651. doi:10.1090/S0002-9939-96-03587-3
[13] F. Karpelevich, “On the Eigenvalues of a Matrix with Nonnegative Elements,” Izv. Akad. Nauk SSSR Ser. Mat. Vol. 15, 1951, pp. 361-383.
[14] R. B. Kellogg, “Matrices Similar to a Positive or Essentially Posi-tive Matrix,” Linear Algebra Applied, Vol. 4, No. 3, 1971, pp. 191-204. doi:10.1016/0024-3795(71)90015-2
[15] C. Knudsen and J. J. McDonald, “A Note on THE Convexity of the Realizable set of Eigenvalues for Nonnegative Symmetric Matrices,” Electronic Journal Linear Algebra, Vol. 8, 2001, pp. 110-114.
[16] T. Laffey, “Realizing Matrices in the Nonnegative Inverse Eigen-value Problem,” Texts in Mathematics ,Series B, University, Coimbra, 1999, pp. 21-32.
[17] T. Laffey and E. Meehan, “A refinement of an inequality of Johnson, Loewy, and London on Nonnegative Matrices and Some Applications,” Electron Journal Linear Algebra, Vol. 3, 1998, pp. 119-128.
[18] T. Laffey and E. Meehan, “A Characterization of Trace zero Nonnegative 5×5 Matrices,” Linear Algebra Applied, Vol. 302-303, No. 1, 1999, pp. 295-302. doi:10.1016/S0024-3795(99)00099-3
[19] J. J. McDonald and M. Neumann, “The Soules Approach to the Inverse Eigenvalue Problem for Nonnegative Symmetric Matrices of Order n-5,” Contemporary Mathematics, Vol. 259, 2000, pp. 387-407.
[20] L. Mirsky and H. Perfect, “Spectral Properties of Doubly Stochastic Matrices,” Mathematics and Statistics, Vol. 69 No. 1, 1965, pp. 35-57. doi:10.1007/BF01313442
[21] H. Perfect, “Methods of Con-structing Certain Stochastic Matrices,” Duke Mathematical Journal, Vol. 20, No. 3, 1953, pp. 395-404. doi:10.1215/S0012-7094-53-02040-7
[22] N. Radwan, “An Inverse Eigenvalue Problem for Symmetric and Normal Matri-ces,” Linear Algebra Applied, Vol. 248, No. 15, 1996, pp. 101-109. doi:10.1016/0024-3795(95)00162-X
[23] R. Reams, “An Ine-quality for Nonnegative Matrices and the Inverse Eigenvalue Problem,” Linear and Multilinear Algebra, Vol. 41, No. 4, 1996, pp. 367-375. doi:10.1080/03081089608818485
[24] G. Wuwen, “Eigen-values of Nonnegative Matrices,” Linear Algebra Applied, Vol. 266, No. 15 1997, pp. 261-270. doi:10.1016/S0024-3795(96)00007-9
[25] X. Z. Zhan, “Matrix Theory,” Academic Press, Chinese, 2008, p. 127.
[26] P. D. Egleston and T. D. Lenker, “Sivaram K. Narayan, the Non-negative Inverse Eigenvalue Problem,” Linear Algebra and Its Applications, Vol. 379, No. 1, 2004, pp. 475-490.

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