Abstract
In this paper we derive new properties complementary to an 2n×2n Brualdi-Li tournament matrix B 2n . We show that B 2n has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of B 2n is also determined. Related results obtained in previous articles are proven to be corollaries.
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References
A. Brauer, I. C. Gentry: Some remarks on tournament matrices. Linear Algebra Appl. 5 (1972), 311–318.
A. Brauer, I. C. Gentry: On the characteristic roots of tournament matrices. Bull. Am. Math. Soc. 74 (1968), 1133–1135.
R. Brualdi, Q. Li: Problem 31. Discrete Mathematics 43 (1983), 329–330.
P. J. Davis: Circulant Matrices. Pure & Applied Mathematics. John Wiley & Sons, New York, 1979.
S. Friedland: Eigenvalues of almost skew symmetric matrices and tournament matrices. Combinatorial and Graph-Theoretical Problems in Linear Algebra (R. A. Brualdi et al., eds. ). Proceedings of a workshop held at the University of Minnesota, USA, 1991, IMA Vol. Math. Appl. 50, Springer, New York, 1993, pp. 189–206.
D. A. Gregory, S. J. Kirkland: Singular values of tournament matrices. Electron. J. Linear Algebra (electronic only) 5 (1999), 39–52.
D. A. Gregory, S. J. Kirkland, B. L. Shader: Pick’s inequality and tournaments. Linear Algebra Appl. 186 (1993), 15–36.
G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 1952.
R. Hemasinha, J. R. Weaver, S. J. Kirkland, J. L. Stuart: Properties of the Brualdi-Li tournament matrix. Linear Algebra Appl. 361 (2003), 63–73.
A. Horn: Doubly stochastic matrices and the diagonal of a rotation matrix. Am. J. Math. 76 (1954), 620–630.
R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1991.
S. Kirkland: An upper bound on the Perron value of an almost regular tournament matrix. Linear Algebra Appl. 361 (2003), 7–22.
S. Kirkland: A note on Perron vectors for almost regular tournament matrices. Linear Algebra Appl. 266 (1997), 43–47.
S. Kirkland: A note on the sequence of Brualdi-Li matrices. Linear Algebra Appl. 248 (1996), 233–240.
S. Kirkland: On the minimum Perron value for an irreducible tournament matrix. Linear Algebra Appl. 244 (1996), 277–304.
S. Kirkland: Hypertournament matrices, score vectors and eigenvalues. Linear Multilinear Algebra 30 (1991), 261–274.
S. J. Kirkland, B. L. Shader: Tournament matrices with extremal spectral properties. Linear Algebra Appl. 196 (1994), 1–17.
J. S. Maybee, N. J. Pullman: Tournament matrices and their generalizations. I. Linear Multilinear Algebra 28 (1990), 57–70.
L. Mirsky: Inequalities and existence theorems in the theory of matrices. J. Math. Anal. Appl. 9 (1964), 99–118.
J. W. Moon: Topics on Tournaments. Holt, Rinehart and Winston, New York, 1968.
J. W. Moon, N. J. Pullman: On generalized tournament matrices. SIAM Rev. 12 (1970), 384–399.
G. Pólya, G. Szegő: Problems and Theorems in Analysis. II Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry. Classics in Mathematics, Springer, Berlin, 1998.
B. L. Shader: On tournament matrices. Linear Algebra Appl. 162–164 (1992), 335–368.
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The research has been supported partly by DIMACS (NSF center at Rutgers, The State University of New Jersey), Shanxi Beiren Jihua Projects of China and The University of Puerto Rico at Mayagüez.
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Wang, C., Yong, X. Some properties complementary to Brualdi-Li matrices. Czech Math J 65, 135–149 (2015). https://doi.org/10.1007/s10587-015-0164-7
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DOI: https://doi.org/10.1007/s10587-015-0164-7