Partly Zero Eigenvectors and Matrices That Satisfy Au=b

For given vectors u; b 2 F n (where F is a eld with at least 3 elements), we establish criteria for deciding whether a digraph allows in its pattern class a matrix A which satisses Au = b: As corollaries to this we give necessary and suucient conditions for a pattern class to allow a matrix which has an eigenvector with a particular zero/nonzero pattern. Moreover we establish whether or not that eigenvector can correspond to a zero or nonzero eigenvalue. We use these results to establish the analogous results for loopfree digraphs, and thus we obtain results additional to work already done by Maybee, Olesky, and Van Den Driessche in this area. We then use bipartite graphs to generalize our criteria for solutions to Au = b to the rectangular case.