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. I n one generalization, which was motivated by economics, the main assumption is that (B - A)(-1) A is nonnegative. In the second generalization, the ma in assumption is that there exists a matrix X greater than or equal to 0 su ch that A = BX. The equivalence of these two assumptions when B is nonsingu lar is considered. For rho(\B(-1)A\) < 1, a complete characterization, invo lving a condition on the digraph of B-1 A, is proved. It is conjectured tha t the characterization holds for p(B-1 A) < 1, and partial results are give n for this case. (C) 1999 Elsevier Science Inc, All rights reserved.