Relations between Perron-Frobenius results for matrix pencils

Two different generalizations of the Perron-Frobenius theory to the matrix pencil Ax = lambda Bx are discussed, and their relationships are studied. In one generalization, which was motivated by economics, the main assumption is that (B - A)(-1) A is nonnegative. In the second generalization, the main assumption is that there exists a matrix X greater than or equal to 0 such that A = BX. The equivalence of these two assumptions when B is nonsingular is considered. For rho(\B(-1)A\) < 1, a complete characterization, involving a condition on the digraph of B-1 A, is proved. It is conjectured that the characterization holds for p(B-1 A) < 1, and partial results are given for this case. (C) 1999 Elsevier Science Inc, All rights reserved.