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The Perron-Frobenius Theorem and Ranking NFL Teams
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The Perron-Frobenius Theorem
and Ranking NFL Teams
Perron- Frobenius Theorem
Definitions
Square Matrix
Eigenvector
If the nontrivial matrix A has nonnegative entries, then there exists an eigenvector with nonnegative entries, corresponding to a positive eigenvalue.
Furthermore, if A is irreducible, the eigenvector has strictly positive entries, is unique and simple, and the corresponding eigenvalue is the largest eigenvalue of A in absolute value.
the multiplier of the eigenvalue, denoted as
New Rankings
Eigenvalues
Nonlinear Scheme:
Non-degenerate Eigenvalue
Direct Method
Nonlinear function of r
compared to:
Assign a score to each team that
is based upon the match:
With any positive vector r0
None of the eigenvalues are equal each other
a non-zero vector v such that when the matrix is multiplied by v, it results in a constant multiple of v
2012-2013 NFL Season
To find the eigenvector:
Actual
Direct Method
Strength of a team should be
proportional to its score:
Patriots
Bengals
Chargers
49ers
Eagles
Lions
Vikings
Packers
Texans
Bears
Cowboys
Ravens
Raiders
Steelers
Buccaneers
Cardinals
Rams
Panthers
Saints
Redskins
Chiefs
Browns
Seahawks
Jets
Colts
Broncos
Doplhins
Bills
Titans
Giants
Falcons
Jaguars
Points should be distributed in
a nonlinear way:
By Lauren Sumereau