On the minimum rank of not necessarily symmetric matrices : a preliminary study

F. Barioli, S.M. Fallat, H.T. Hall, D. Hershkowitz, L. Hogben, H. Holst, van der, B.L. Shader

Research output: Contribution to journalArticleAcademicpeer-review

30 Citations (Scopus)
132 Downloads (Pure)

Abstract

The minimum rank of a directed graph G is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i _= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.
Original languageEnglish
Pages (from-to)126-145
JournalElectronic Journal of Linear Algebra
Volume18
Publication statusPublished - 2009

Fingerprint

Dive into the research topics of 'On the minimum rank of not necessarily symmetric matrices : a preliminary study'. Together they form a unique fingerprint.

Cite this