This is the second of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses Keener’s method (see J.P. Keener, *The Perron-Frobenius theorem and the ranking of football,* SIAM Review 35 (1993)80-93 for details).

See the first post in the series for a discussion of the data we’re using to explain this method. We recall the table of results:

XY | Army | Bucknell | Holy Cross | Lafayette | Lehigh | Navy |

Army | x | 14-16 | 14-13 | 14-24 | 10-12 | 8-19 |

Bucknell | 16-14 | x | 27-30 | 18-16 | 23-20 | 10-22 |

Holy Cross | 13-14 | 30-27 | x | 19-15 | 17-13 | 9-16 |

Lafayette | 24-14 | 16-18 | 15-19 | x | 12-23 | 17-39 |

Lehigh | 12-10 | 20-23 | 13-17 | 23-12 | x | 12-18 |

Navy | 19-8 | 22-10 | 16-9 | 39-17 | 18-12 | x |

Win-loss digraph of the Patriot league mens baseball from 2015

Suppose T teams play each other. Let $latex A=(a_{ij})_{1leq i,jleq T}$ be a non-negative square matrix determined by the results of their games, called the *preference matrix*

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