Tournament Matrix

A matrix for a round-robin tournament involving  players competing in  matches (no ties allowed) having entries

(1)

This scoring system differs from that used to compute a score sequence of a tournament, in which a win gives one point and a loss zero points. The matrix satisfies

(2)

where  is the transpose of  (McCarthy and Benjamin 1996).

The tournament matrix for  players has zero determinant iff  is odd (McCarthy and Benjamin 1996). Furthermore, the dimension of the null space of an -player tournament matrix is

(3)

(McCarthy and Benjamin 1996).

REFERENCES

McCarthy, C. A. and Benjamin, A. T. "Determinants of the Tournaments." Math. Mag. 69, 133-135, 1996.

Michael, T. S. "The Ranks of Tournament Matrices." Amer. Math. Monthly 102, 637-639, 1995.