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Open problems on GKK \(\tau\)-matrices. (English) Zbl 1160.15305

Summary: We propose several open problems on GKK \(\tau\)-matrices raised by examples showing that some such matrices are unstable.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A29 Inverse problems in linear algebra
15B57 Hermitian, skew-Hermitian, and related matrices
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[1] Biegler-König, F. W., Construction of band matrices from spectral data, Linear Algebra Appl., 40, 79-87 (1981) · Zbl 0468.15006
[2] Carlson, D., Weakly sign-symmetric matrices and some determinantal inequalities, Colloq. Math., 17, 123-129 (1967) · Zbl 0147.27502
[3] Carlson, D., A class of positive stable matrices, J. Res. Nat. Bur. Standards B, 78, 1-2 (1974) · Zbl 0281.15020
[4] N.G. Čebotarev, N.N. Meı̆man, The Routh-Hurwitz problem for polynomials and entire functions. Real Quasipolynomials with \(r=3,\); N.G. Čebotarev, N.N. Meı̆man, The Routh-Hurwitz problem for polynomials and entire functions. Real Quasipolynomials with \(r=3,\)
[5] Engel, G. M.; Schneider, H., The Hadamard-Fischer inequality for a class of matrices defined by eigenvalue monotonicity, Linear and Multilinear Algebra, 4, 155-176 (1976)
[6] Fan, K., Subadditive functions on a distributive lattice and an extension of Szász’s inequality, J. Math. Anal. Appl., 18, 262-268 (1967) · Zbl 0204.02701
[7] Gantmacher, F. R.; Krein, M. G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (1950), Gostechizdat · Zbl 0041.35502
[8] Hershkowitz, D., Recent directions in matrix stability, Linear Algebra Appl., 171, 161-186 (1992) · Zbl 0759.15010
[9] Hershkowitz, D.; Berman, A., Notes on \(ω\)- and \(τ\)-matrices, Linear Algebra Appl., 58, 169-183 (1984) · Zbl 0543.15014
[10] Kotelyansky (Koteljanskii), D. M., A property of sign-symmetric matrices, Uspekhi Mat. Nauk. (NS), 8, 163-167 (1952), also Amer. Math. Soc. Transl. Ser. 2, 27 (1963) 19-23
[11] Holtz, O., Not all GKK \(τ\)-matrices are stable, Linear Algebra Appl., 291, 235-244 (1999) · Zbl 0968.15014
[12] Mehrmann, V., On some conjectures on the spectra of \(τ\)-matrices, Linear and Multilinear Algebra, 16, 101-112 (1984) · Zbl 0574.15007
[13] C. Niculescu, A new look at Newton’s inequalities, J. Inequal. Pure Appl. Math. 1 (2) (2000); C. Niculescu, A new look at Newton’s inequalities, J. Inequal. Pure Appl. Math. 1 (2) (2000) · Zbl 0972.26010
[14] Varga, R., Recent results in linear algebra and its applications, (in Numerical Methods in: Linear Algebra, Proceedings of the Third Seminar of Numerical Applied Mathematics. in Numerical Methods in: Linear Algebra, Proceedings of the Third Seminar of Numerical Applied Mathematics, Akad. Nauk. SSSR Sibirsk, Otdel. Vychisl. Tsentr. Novosibirsk (1978)), 5-15 · Zbl 0435.15002
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