The Boolean pivot operation M-matrices, and reducible matrices

Abstract

Let A be a Boolean n × n matrix, and let G = ( N, U) be the corresponding diagraph, where N ≔ {1,…, n} is the set of vertices and U ⊂ N × N is the set of arcs of G. For any R ⊂ N, combining the steps r ϵ R of Warshall's algorithm for determining the reachability matrix of A yields the Boolean pivot operation B R. The matrix Ā ≔ B R A is the so-called R-reachability matrix of A: a ̄ ij = 1 if and only if a ij = 1 or there is a connection between i and j via vertices belonging to R. We also have B R B S = B S B R = B R∪S for any R, S ⊂ N. The Boolean pivot operation is closely related to the principal pivotal operation for real matrices. So we obtain the Boolean analogues to the formula for inverting a real block matrix and for the Sherman-Morrison-Woodbury formulae for updating the inverse of a real matrix. Using the close connection between nonsingular M-matrices and the corresponding Boolean matrices, we obtain a flexible algorithm for excluding and including vertices one at a time in G while retaining the original connections between the vertices of the current diagraph. We derive some criteria for irreducibility of Boolean block matrices and of partitioned Z-matrices and give a condensed form of Warshall's algorithm for testing whether a Boolean matrix is irreducible. Moreover, we correct two recent results by R.L. Smith on irreducibility of real block matrices.