A scalar matching factor on the Birkhoff polytope characterizing permutation and uniform matrices

TL;DR

This paper introduces a scalar matching factor on the Birkhoff polytope, which characterizes permutation and uniform matrices by their extremal properties, and extends this concept to a broader class of matrices.

Contribution

It defines a new scalar quantity called the matching factor on the Birkhoff polytope and demonstrates its extremal properties for permutation and uniform matrices, extending to larger matrix classes.

Findings

Permutation matrices maximize the matching factor.

Uniform matrices minimize the matching factor.

The matching factor extends to larger matrix classes with similar extremal properties.

Abstract

Birkhoff polytope is the set of all bistochastic matrices (also known as doubly stochastic matrices). Bistochastic matrices form a special class of stochastic matrices where each row and column sums up to one. Permutation matrices and uniform matrices are special extreme cases of bistochastic matrices. In this paper, we define a scalar quantity called the matching factor on the Birkhoff polytope. Given a bistochastic matrix, we define the matching factor by taking the product of the squares of the Euclidean norms of each row and column and show that permutation matrices and uniform matrices maximize and minimize the matching factor, respectively. We also extend this definition of scalar matching factor to a larger class of matrices and show similar maximization and minimization properties.