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).