Reconstructing matrices from minors

Ashlyn M. Winkler
Joshua J. Mollner
Charles R. Johnson, William & Mary

Abstract

Which collections of mn minors of an m-by-n matrix uniquely determine the matrix, given some regularity conditions? For m = n = 3, the 585 such collections, that are distinct up to symmetry, are determined. For general m, n, a necessary and a sufficient condition for reconstruction are given in terms of matchings in a bipartite graph. Among other particular results, those collections of entries for which there are minors that permit reconstruction one entry at a time are characterized. (C) 2010 Elsevier Inc. All rights reserved.