A matrix version of the Steinitz lemma

TL;DR

This paper extends the classical Steinitz lemma to a matrix setting, providing a tighter bound on the rearrangement of matrix entries to control the norm of partial sums.

Contribution

It introduces a matrix version of the Steinitz lemma and improves the bound on the norm of partial sums from previous results.

Findings

Rearrangement of matrix entries with controlled partial sum norms

Improved bound from 40d^5 to (4d-2) times the maximum entry norm

Enhanced understanding of vector rearrangements in matrix contexts

Abstract

The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the matrix version $A$ is a $k\times n$ matrix with entries $a_i^j \in \R^d$ with $\sum_{j=1}^k\sum_{i=1}^na_i^j=0$. It is proved in \cite{OPW} that there is a rearrangement of row $j$ of $A$ (for every $j$) such that the sum of the entries in the first $m$ columns of the rearranged matrix has norm at most $40d^5\max \|a_i^j\|$ (for every $m$). We improve this bound to $(4d-2)\max \|a_i^j\|$.