Tournament Matrix

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

$$a_{ij}=\cases{ 1 & if player $i$\ defeats player $j$\cr -1 & if player $i$\ loses to player $j$\cr 0 & if $i=j$.\cr}$$

The Matrix satisfies

$${\hbox{\sf A}}+{\hbox{\sf A}}^{\rm T}+{\hbox{\sf I}}={\hbox{\sf J}},$$

where ${\hbox{\sf I}}$ is the Identity Matrix, ${\hbox{\sf J}}$ is an $n\times n$ Matrix of all 1s, and ${\hbox{\sf A}}^{\rm T}$ is the Matrix Transpose of A.

The tournament matrix for $n$ players has zero Determinant Iff $n$ is Odd (McCarthy and Benjamin 1996). The dimension of the Nullspace of an $n$-player tournament matrix is

$$\mathop{\rm dim}{\rm [nullspace]}=\cases{ 0 & for $n$\ even\cr 1 & for $n$\ odd\cr}$$

(McCarthy 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.