Geodesic
Even and odd tournament matrices with minimum rank over finite fields
The electronic journal of linear algebra, Tome 22 (2011), pp. 363-377.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The (0, 1)-matrix A of order n is a tournament matrix provided A + A T+ I = J, where I is the identity matrix, and J = J n is the all 1's matrix of order n. It was shown by de Caen and Michael that the rank of a tournament matrix A of order n over a field of characteristic p satisfies rank p (A) $\geq $(n - 1)/2 with equality if and only if n is odd and AA T= O. This article shows that the rank of a tournament matrix A of even order n over a field of characteristic p satisfies rank p (A) $\geq $n/2 with equality if and only if after simultaneous row and column permutations AA $T= \pm J$ m OO O, for a suitable integer m. The results and constructions for even order tournament matrices are related to and shed light on tournament matrices of odd order with minimum rank.
Classification : 15A03, 05C20, 05C50
Keywords: tournament matrix, rank 
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     author = {Doering, Elizabeth and Michael, T.S. and Shader, Bryan L.},
     title = {Even and odd tournament matrices with minimum rank over finite fields},
     journal = {The electronic journal of linear algebra},
     pages = {363--377},
     publisher = {mathdoc},
     volume = {22},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2011__22__a56/}
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Doering, Elizabeth; Michael, T.S.; Shader, Bryan L. Even and odd tournament matrices with minimum rank over finite fields. The electronic journal of linear algebra, Tome 22 (2011), pp. 363-377. http://geodesic.mathdoc.fr/item/ELA_2011__22__a56/