In the literature some interesting in-depth analysis on possible ranking methods, mainly concentrated on football teams ranking, may be found. Within certain hypotheses the ranking problem may be easily formulated as a linear eigenvalue problem. In this paper we present the application of the linear model to the recent editions of the Italian national soccer championship, showing that the results from the model give some significant differences with respect to the final official ordering. We recall first some important mathematical definitions and results, which are relevant in the linear model. Among these, the concepts of eigenvalues and eigenvectors of a square matrix, together with the fundamental properties of the eigenvalues of positive matrices, namely Perron–Frobenius theorem on the so called dominant eigenvalue. Then we collect the data, mainly the final results of the matches played by the teams in the Italian championship, in order to have the matrix of the model. The computation of the dominant eigenvalue and the corresponding eigenvector gives us the solution of the model, that is a final ordering for the teams in the championship. Finally we try to interpret the result, by comparing this to the official final ordering.

### A linear model for ranking soccer teams

#### Abstract

In the literature some interesting in-depth analysis on possible ranking methods, mainly concentrated on football teams ranking, may be found. Within certain hypotheses the ranking problem may be easily formulated as a linear eigenvalue problem. In this paper we present the application of the linear model to the recent editions of the Italian national soccer championship, showing that the results from the model give some significant differences with respect to the final official ordering. We recall first some important mathematical definitions and results, which are relevant in the linear model. Among these, the concepts of eigenvalues and eigenvectors of a square matrix, together with the fundamental properties of the eigenvalues of positive matrices, namely Perron–Frobenius theorem on the so called dominant eigenvalue. Then we collect the data, mainly the final results of the matches played by the teams in the Italian championship, in order to have the matrix of the model. The computation of the dominant eigenvalue and the corresponding eigenvector gives us the solution of the model, that is a final ordering for the teams in the championship. Finally we try to interpret the result, by comparing this to the official final ordering.
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2019
Ranking scheme, Linear transformation, Eigenvalues, Dominant eigenvalue
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