Realizability criterion for the symmetric nonnegative inverse eigenvalue problem. (English) Zbl 1097.15013
A set \(\Lambda\) of complex numbers is said to be realizable if \(\Lambda\) is the spectrum of an entrywise nonnegative matrix. Motivated by A. Brauer’s theorem [Duke Math. J. 19, 75–91 (1952; Zbl 0046.01202)], in an earlier paper [Linear Algebra Appl. 369, 169–184 (2003; Zbl 1031.15018)] the author showed how to explicitly construct nonnegative matrices realizing the prescribed real spectrum. Here the sufficiency of that realizability criterion is proved for the existence of a symmetric nonnegative matrix with prescribed spectrum satisfying some conditions. Two illustrative examples are provided.
Reviewer: C. M. da Fonseca (Coimbra)
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A29 | Inverse problems in linear algebra |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
Keywords:
symmetric nonnegative inverse eigenvalue problem; nonnegative matrices; realizability criterion; prescribed spectrumReferences:
| [1] | Borobia, A., On the nonnegative eigenvalue problem, Linear Algebra Appl., 223-224, 131-140 (1995) · Zbl 0831.15014 |
| [2] | Borobia, A.; Moro, J.; Soto, R., Negativity compensation in the nonnegative inverse eigenvalue problem, Linear Algebra Appl., 393, 73-89 (2004) · Zbl 1079.15503 |
| [3] | Brauer, A., Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices, Duke Math. J., 19, 75-91 (1952) · Zbl 0046.01202 |
| [4] | Egleston, P.; Lenker, T.; Narayan, S., The nonnegative inverse eigenvalue problem, Linear Algebra Appl., 379, 475-490 (2004) · Zbl 1040.15009 |
| [5] | Fiedler, M., Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl., 9, 119-142 (1974) · Zbl 0322.15015 |
| [6] | Johnson, C. R., Row stochastic matrices similar to doubly stochastic matrices, Linear and Multilinear Algebra, 10, 113-130 (1981) · Zbl 0455.15019 |
| [7] | Johnson, C. R.; Laffey, T. J.; Loewy, R., The real and the symmetric nonnegative inverse eigenvalue problems are different, Proc. AMS, 124, 3647-3651 (1996) · Zbl 0861.15007 |
| [8] | Kellogg, R., Matrices similar to a positive or essentially positive matrix, Linear Algebra Appl., 4, 191-204 (1971) · Zbl 0215.37504 |
| [9] | Laffey, T. J.; Meehan, E., A characterization of trace zero nonnegative 5×5 matrices, Linear Algebra Appl., 302-303, 295-302 (1999) · Zbl 0946.15008 |
| [10] | Loewy, R.; London, D., A note on an inverse problem for nonnegative matrices, Linear and Multilinear Algebra, 6, 83-90 (1978) · Zbl 0376.15006 |
| [11] | Mcdonald, J. J.; Newmann, M., The Soules approach to the inverse eigenvalue problem for nonnegative symmetric matrices of order \(n\)⩽5, Contemp. Math., 259, 387-407 (2000) · Zbl 0965.15009 |
| [12] | Minc, H., Nonnegative Matrices (1988), John Wiley & Sons: John Wiley & Sons New York · Zbl 0638.15008 |
| [13] | Perfect, H., Methods of constructing certain stochastic matrices, Duke Math. J., 20, 395-404 (1953) · Zbl 0051.35701 |
| [14] | Perfect, H., Methods of constructing certain stochastic matrices II, Duke Math. J., 22, 305-311 (1955) · Zbl 0068.32704 |
| [15] | Radwan, N., An inverse eigenvalue problem for symmetric and normal matrices, Linear Algebra Appl., 248, 101-109 (1996) · Zbl 0865.15008 |
| [16] | Reams, R., An inequality for nonnegative matrices and the inverse eigenvalue problem, Linear and Multilinear Algebra, 41, 367-375 (1996) · Zbl 0887.15015 |
| [17] | Salzmann, F., A note on eigenvalues of nonnegative matrices, Linear Algebra Appl., 5, 329-338 (1972) · Zbl 0255.15009 |
| [18] | Šmigoc, H., The inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl., 393, 365-374 (2004) · Zbl 1075.15012 |
| [19] | Soto, R., Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra Appl., 369, 169-184 (2003) · Zbl 1031.15018 |
| [20] | Soto, R.; Borobia, A.; Moro, J., On the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem, Linear Algebra Appl., 396, 223-241 (2005) · Zbl 1086.15010 |
| [21] | Soto, R., Realizability by symmetric nonnegative matrices, Proyecciones J. Math., 24, 65-78 (2005) · Zbl 1097.15012 |
| [22] | Soules, G., Constructing symmetric nonnegative matrices, Linear and Multilinear Algebra, 13, 241-251 (1983) · Zbl 0516.15013 |
| [23] | Suleimanova, H. R., Stochastic matrices with real characteristic values, Dokl. Akad. Nauk SSSR, 66, 343-345 (1949) · Zbl 0035.20903 |
| [24] | Wuwen, G., An inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl., 249, 67-78 (1996) · Zbl 0865.15009 |
| [25] | Wuwen, G., Eigenvalues of nonnegative matrices, Linear Algebra Appl., 266, 261-270 (1997) · Zbl 0903.15003 |
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