Tournament Matrix

A matrix for a round-robin tournament involving n players competing in n(n-1)/2 matches (no ties allowed) having entries

 a_(ij)={1   if player i defeats player j; -1   if player i loses to player j; 0   if i=j.

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


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

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

 dim[nullspace]={0   for n even; 1   for n odd

(McCarthy and Benjamin 1996).

See also


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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.

Referenced on Wolfram|Alpha

Tournament Matrix

Cite this as:

Weisstein, Eric W. "Tournament Matrix." From MathWorld--A Wolfram Web Resource.

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