  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 $(4d-2)\max
\|a_i^j\|$.
