# Matrix scaling and explicit doubly stochastic limits

## TL;DR

This paper derives exact formulas for the Sinkhorn limits of specific symmetric positive 3x3 matrices, enhancing understanding of matrix scaling convergence to doubly stochastic matrices.

## Contribution

It provides explicit formulas for Sinkhorn limits of certain symmetric 3x3 matrices, a novel contribution to matrix scaling theory.

## Key findings

- Exact formulas for Sinkhorn limits of specific matrices
- Enhanced understanding of convergence in matrix scaling
- Explicit characterization of symmetric 3x3 cases

## Abstract

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, often called the \emph{Sinkhorn limit} of $A$. The main result in this paper is the computation of exact formulae for the Sinkhorn limits of certain symmetric positive $3\times 3$ matrices.