The WST-decomposition for partial matrices

TL;DR

This paper introduces the WST-decomposition, a unique block matrix form for partial matrices that simplifies the computation of maximum rank completions and extends to a broader class called ACI-matrices.

Contribution

The paper presents the WST-decomposition, a novel, unique block decomposition for partial matrices that facilitates rank analysis and extends to ACI-matrices.

Findings

WST-decomposition provides a straightforward way to compute maximum rank completions.

The decomposition is unique up to elementary row operations and column permutation.

It applies to a broader class of matrices called ACI-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.