# Matrix scaling, explicit Sinkhorn limits, and arithmetic

## TL;DR

This paper explores the convergence of matrix scaling to doubly stochastic matrices, providing explicit formulas for certain symmetric 3x3 matrices and connecting the results to diophantine approximation.

## Contribution

It offers explicit formulas for Sinkhorn limits of specific symmetric 3x3 matrices and links matrix scaling to diophantine approximation problems.

## Key findings

- Explicit formulas for Sinkhorn limits of symmetric 3x3 matrices.
- Connections established between matrix scaling and diophantine approximation.
- Analysis of convergence properties in matrix scaling processes.

## 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 $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn limits of certain symmetric positive $3\times 3$ matrices are computed, and related problems in diophantine approximation are considered.