Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T02:00:45.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Pseudo-Tournament Matrices and Their Eigenvalues

Published online by Cambridge University Press:  28 May 2015

Chuanlong Wang  and
Xuerong Yong
Chuanlong Wang*
Affiliation:
Department of Mathematics, Taiyuan Normal University, Taiyuan, Shanxi, China
Xuerong Yong*
Affiliation:
Department of Mathematical Sciences, The University of Puerto Rico, Mayaguez, PR 00681, USA
*
Corresponding author. Email address: [email protected]
Corresponding author. Email address: [email protected]
Get access

Abstract

A tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. An n × n complex matrix A is called h-pseudo-tournament if there exists a complex or real nonzero column vector h such that A + A* = hh* − I. This class of matrices is a generalisation of well-studied tournament-like matrices such as h-hypertournament matrices, generalised tournament matrices, tournament matrices, and elliptic matrices. We discuss the eigen-properties of an h-pseudo-tournament matrix, and obtain new results when the matrix specialises to one of these tournament-like matrices. Further, several results derived in previous articles prove to be corollaries of those reached here.

Keywords

15A1505C20Pseudo-tournament matrixeigenvaluespectral radiustournament matrix
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 3 , August 2014 , pp. 205 - 221
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Brauer, A. and Gentry, I., On the characteristic roots of tournament matrices, Bull. Amer. Math. Soc. 74, 1133–1135 (1968).CrossRefGoogle Scholar
[2]Brauer, A. and Gentry, I., Some remarks on tournament matrices, Linear Algebra Appl. 5, 311–318 (1972).Google Scholar
[3]de Caen, D., Gregory, D.A., Kirkland, S.J., Pullman, N.J. and Maybee, J.S., Algebraic multiplicity of the eigenvalues of a tournament matrix, Linear Algebra Appl. 169, 179–193 (1992).Google Scholar
[4]Fiedler, M., Elliptic matrices with zero diagonal, Linear Algebra Appl. 197–198, 337–347 (1994).CrossRefGoogle Scholar
[5]Gregory, D.A. and Kirkland, S.J., Singular values of the tournament matrices, The Electronic Journal of Linear Algebra 5, 39–52 (1999).Google Scholar
[6]Gregory, D., Kirkland, S. and Shader, B., Pick inequality and tournaments, Linear Algebra Appl. 186, 15–36 (1993).CrossRefGoogle Scholar
[7]Hardy, G.H., Littlewood, J.E. and Polya, G., Inequalities, 2nd edition. Cambridge University Press, 1952.Google Scholar
[8]Horn, A., Doubly stochastic matrices and the diagonals of a rotation matrix, Amer. J. Math. 76, 620–630 (1954).Google Scholar
[9]Horn, R. and Johnson, C.R., Matrix Analysis. Cambridge University Press, 1991.CrossRefGoogle Scholar
[10]Kirkland, S., Hyper tournament matrices, score vectors and eigenvalues, Linear and Multilinear Algebra 30, 261–274 (1991).CrossRefGoogle Scholar
[11]Kirkland, S. and Shader, B., Tournament matrices with extremal spectral properties, Linear Algebra Appl. 196, 1–17 (1994).CrossRefGoogle Scholar
[12]Hemasinha, R., Weaver, J., Kirkland, S. and Stuart, J., Properties of the Brualdi-Li tournament matrix, Linear Algebra Appl. 361, 63–73 (2003).CrossRefGoogle Scholar
[13]Maybee, S. and Pullman, N., Tournament matrices and their generalisations, Linear and Mulilinear Algebra 28, 57–70 (1990).Google Scholar
[14]Mirsky, L., Inequalities and existence theorems in the theory of matrices, J. Math. Anal. Appl. 9, 99–118 (1964).Google Scholar
[15]Moon, J., Topics on Tournaments. Holt, Reinhart and Winston, New York, 1968.Google Scholar
[16]Moon, J. and Pullman, N., On generalized tournaments, SIAM Review 12, 384–399 (1970).CrossRefGoogle Scholar
[17]Ostrowski, A. and Schneider, H., Some theorems on the interia of general matrices, J. Math. Anal. Appl. 4, 72–84 (1962).Google Scholar
[18]Shader, B., On tournament matrices, Linear Algebra Appl. 162–164, 335–368 (1992).Google Scholar
[19]Yong, X., Elliptic matrices and their eigenpolynomials, Linear Algebra Appl. 259, 347–356 (1997).CrossRefGoogle Scholar
[20]Yong, X., On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295, 73–80 (1999).Google Scholar