The WST-decomposition for partial matrices

Abstract

A partial matrix over a field F is a matrix whose entries are either an element of F or an indeterminate, and each indeterminate only appears once. A completion is a matrix obtained by giving values to each of the indeterminates. We can decompose any partial matrix, through elementary row operations and column permutations, into a block matrix of the form [W⁎⁎0S⁎00T] where W is wide (has more columns than rows), S is square, T is tall (has more rows than columns), and with these three blocks having at least one completion with full rank. And importantly, W, S and T are unique up to elementary row operations and column permutations whenever S is required to be as large as possible. When this is the case, [W⁎⁎0S⁎00T] will be called a WST-decomposition. As an application of this decomposition, we will see that the maximal rank over all completions of the original partial matrix is rows(W)+rows(S)+cols(T). In fact, we introduce the WST-decomposition for a broader class of matrices: the ACI-matrices.