Tournament Matrix
المؤلف:
McCarthy, C. A. and Benjamin, A. T.
المصدر:
"Determinants of the Tournaments." Math. Mag. 69
الجزء والصفحة:
...
14-4-2022
1910
Tournament Matrix
A matrix for a round-robin tournament involving 
players competing in 
matches (no ties allowed) having entries
![a_(ij)=<span style=]() {1 if player i defeats player j; -1 if player i loses to player j; 0 if i=j. " src="https://mathworld.wolfram.com/images/equations/TournamentMatrix/NumberedEquation1.svg" style="height:81px; width:266px" /> |
(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
![dim[nullspace]=<span style=]() {0 for n even; 1 for n odd " src="https://mathworld.wolfram.com/images/equations/TournamentMatrix/NumberedEquation3.svg" style="height:53px; width:230px" /> |
(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.
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