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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

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