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A note on the sequence of Brualdi-Li matrices. (English) Zbl 0865.15014

Author’s abstract: We consider the sequence \(2n(n- {1\over 2} -\rho_{2n})\), where \(\rho_{2n}\) is the Perron value of the Brualdi-Lie matrix of order \(2n\) [cf. R. A. Brualdi and Q. Li, Problem 31, Discrete Math. 43, 329-330 (1983)]. We prove that the sequence converges, find its limit, and show that asymptotically, the sequence is monotonically decreasing. In particular, this addresses some problems raised by S. Friedland and M. Katz [Linear Algebra Appl. 208-209, 455-469 (1994; Zbl 0807.15007)].

MSC:

15B36 Matrices of integers
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 0807.15007
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References:

[1] Brauer, A.; Gentry, I., On the characteristic roots of tournament matrices, Bull. Amer. Math. Soc., 74, 1133-1135 (1968) · Zbl 0167.03002
[2] Brualdi, R. A.; Li, Q., Problem 31, Discrete Math., 43, 329-330 (1983)
[3] Friedland, S., Eigenvalues of almost skew-symmetric matrices and tournament matrices, (Brauldi, R. A.; Friedland, S.; Klee, V., Combinatorial and Graph Theoretic Problems in Linear Algebra. Combinatorial and Graph Theoretic Problems in Linear Algebra, IMA Vol. Math. Appl., 50 (1993), Springer-Verlag: Springer-Verlag New York), 189-206 · Zbl 0789.15019
[4] Friedland, S.; Katz, M., On the maximal spectral radius of even tournament matrices and the spectrum of almost skew-symmetric compact operators, Linear Algebra Appl., 208, 455-469 (1994) · Zbl 0807.15007
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