  A partial matrix over a field $\mathbb{F}$ is a matrix whose entries are
either an element of $\mathbb{F}$ or an indeterminate and with each
indeterminate only appearing once. A completion is an assignment of values in
$\mathbb{F}$ to all indeterminates. Given a partial matrix, through elementary
row operations and column permutation it can be decomposed into a block matrix
of the form $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 &
0 & {\bf T} \end{smallmatrix}\right]$ where ${\bf W}$ is wide (has more columns
than rows), ${\bf S}$ is square, ${\bf T}$ is tall (has more rows than
columns), and these three blocks have at least one completion with full rank.
And importantly, each one of the blocks ${\bf W}$, ${\bf S}$ and ${\bf T}$ is
unique up to elementary row operations and column permutation whenever ${\bf
S}$ is required to be as large as possible. When this is the case
$\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T}
\end{smallmatrix}\right]$ will be called a WST-decomposition. With this
decomposition it is trivial to compute maximum rank of a completion of the
original partial matrix: $\#\mbox{rows}({\bf W})+\#\mbox{rows}({\bf
S})+\#\mbox{cols}({\bf T})$. In fact we introduce the WST-decomposition for a
broader class of matrices: the ACI-matrices.
