# Matrix scaling limits in finitely many iterations

## TL;DR

This paper studies the convergence behavior of the Sinkhorn scaling algorithm, providing explicit examples of matrices that become doubly stochastic after exactly one column scaling, highlighting specific limits of the iterative process.

## Contribution

It constructs a two-parameter family of positive matrices that reach doubly stochastic form after a single column scaling, revealing new insights into the algorithm's convergence properties.

## Key findings

- Matrices that become doubly stochastic after one column scaling
- Explicit construction of such matrices for any size n
- Insights into the limits of the Sinkhorn scaling algorithm

## Abstract

The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter family of row but not column stochastic $n\times n$ positive matrices is constructed that become doubly stochastic after exactly one column scaling.