| Title:
|
Some properties complementary to Brualdi-Li matrices (English) |
| Author:
|
Wang, Chuanlong |
| Author:
|
Yong, Xuerong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
135-149 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we derive new properties complementary to an $2n \times 2n$ 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. (English) |
| Keyword:
|
tournament matrix |
| Keyword:
|
Brualdi-Li matrix |
| Keyword:
|
eigenvalue |
| Keyword:
|
Perron value |
| MSC:
|
05C20 |
| MSC:
|
05C50 |
| MSC:
|
15A15 |
| idZBL:
|
Zbl 06433725 |
| idMR:
|
MR3336029 |
| DOI:
|
10.1007/s10587-015-0164-7 |
| . |
| Date available:
|
2015-04-01T12:27:50Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144217 |
| . |
| Reference:
|
[1] Brauer, A., Gentry, I. C.: Some remarks on tournament matrices.Linear Algebra Appl. 5 (1972), 311-318. Zbl 0246.15012, MR 0304206 |
| Reference:
|
[2] Brauer, A., Gentry, I. C.: On the characteristic roots of tournament matrices.Bull. Am. Math. Soc. 74 (1968), 1133-1135. Zbl 0167.03002, MR 0232784, 10.1090/S0002-9904-1968-12079-8 |
| Reference:
|
[3] Brualdi, R., Li, Q.: Problem 31.Discrete Mathematics 43 (1983), 329-330. |
| Reference:
|
[4] Davis, P. J.: Circulant Matrices. Pure & Applied Mathematics.John Wiley & Sons New York (1979). MR 0543191 |
| Reference:
|
[5] Friedland, S.: Eigenvalues of almost skew symmetric matrices and tournament matrices.Combinatorial and Graph-Theoretical Problems in Linear Algebra; Proceedings of a workshop held at the University of Minnesota, USA, 1991; IMA Vol. Math. Appl. 50 Springer, New York (1993), 189-206 R. A. Brualdi et al. Zbl 0789.15019, MR 1240964 |
| Reference:
|
[6] Gregory, D. A., Kirkland, S. J.: Singular values of tournament matrices.Electron. J. Linear Algebra (electronic only) 5 (1999), 39-52. Zbl 0915.05085, MR 1668927 |
| Reference:
|
[7] Gregory, D. A., Kirkland, S. J., Shader, B. L.: Pick's inequality and tournaments.Linear Algebra Appl. 186 (1993), 15-36. Zbl 0776.05072, MR 1217195 |
| Reference:
|
[8] Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities.Cambridge Mathematical Library Cambridge Univ. Press, Cambridge (1952). Zbl 0047.05302, MR 0046395 |
| Reference:
|
[9] Hemasinha, R., Weaver, J. R., Kirkland, S. J., Stuart, J. L.: Properties of the Brualdi-{L}i tournament matrix.Linear Algebra Appl. 361 (2003), 63-73. Zbl 1017.05074, MR 1955553 |
| Reference:
|
[10] Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix.Am. J. Math. 76 (1954), 620-630. Zbl 0055.24601, MR 0063336, 10.2307/2372705 |
| Reference:
|
[11] Horn, R. A., Johnson, C. R.: Matrix Analysis.Cambridge University Press Cambridge (1991). Zbl 0729.15001 |
| Reference:
|
[12] Kirkland, S.: An upper bound on the Perron value of an almost regular tournament matrix.Linear Algebra Appl. 361 (2003), 7-22. Zbl 1019.15004, MR 1955551 |
| Reference:
|
[13] Kirkland, S.: A note on Perron vectors for almost regular tournament matrices.Linear Algebra Appl. 266 (1997), 43-47. Zbl 0901.15011, MR 1473192 |
| Reference:
|
[14] Kirkland, S.: A note on the sequence of Brualdi-{L}i matrices.Linear Algebra Appl. 248 (1996), 233-240. Zbl 0865.15014, MR 1416458, 10.1016/0024-3795(95)00196-4 |
| Reference:
|
[15] Kirkland, S.: On the minimum Perron value for an irreducible tournament matrix.Linear Algebra Appl. 244 (1996), 277-304. Zbl 0860.15016, MR 1403286 |
| Reference:
|
[16] Kirkland, S.: Hypertournament matrices, score vectors and eigenvalues.Linear Multilinear Algebra 30 (1991), 261-274. Zbl 0751.15009, MR 1129183, 10.1080/03081089108818111 |
| Reference:
|
[17] Kirkland, S. J., Shader, B. L.: Tournament matrices with extremal spectral properties.Linear Algebra Appl. 196 (1994), 1-17. Zbl 0790.15021, MR 1273972 |
| Reference:
|
[18] Maybee, J. S., Pullman, N. J.: Tournament matrices and their generalizations. I.Linear Multilinear Algebra 28 (1990), 57-70. Zbl 0714.05041, MR 1077735, 10.1080/03081089008818030 |
| Reference:
|
[19] Mirsky, L.: Inequalities and existence theorems in the theory of matrices.J. Math. Anal. Appl. 9 (1964), 99-118. Zbl 0133.26202, MR 0163918, 10.1016/0022-247X(64)90009-5 |
| Reference:
|
[20] Moon, J. W.: Topics on Tournaments.Holt, Rinehart and Winston New York (1968). Zbl 0191.22701, MR 0256919 |
| Reference:
|
[21] Moon, J. W., Pullman, N. J.: On generalized tournament matrices.SIAM Rev. 12 (1970), 384-399. Zbl 0198.03804, MR 0272644, 10.1137/1012081 |
| Reference:
|
[22] Pólya, G., Szegő, G.: Problems and Theorems in Analysis. II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry.Classics in Mathematics Springer, Berlin (1998). Zbl 1024.00003, MR 1492448 |
| Reference:
|
[23] Shader, B. L.: On tournament matrices.Linear Algebra Appl. 162-164 (1992), 335-368. Zbl 0744.15015, MR 1148408 |
| . |
