The WST-decomposition for partial matrices

Abstract

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.