Diagonal Sums of Doubly Stochastic Matrices

TL;DR

This paper characterizes doubly stochastic matrices where all diagonals avoiding zeros have equal sums, providing insights into their structure and pattern classes.

Contribution

It offers a new characterization of matrices in _n with equal-sum diagonals avoiding zeros, expanding understanding of their structural properties.

Findings

Characterization of matrices with equal-sum diagonals avoiding zeros

Identification of classes of such matrices based on patterns

Theoretical framework for analyzing diagonal sums in doubly stochastic matrices

Abstract

Let $\Omega_n$ denote the class of $n \times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in $\Omega_n$. The main question is: which $A \in \Omega_n$ are such that the diagonals in $A$ that avoid the zeros of $A$ all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.