Relations between Perron—Frobenius results for matrix pencils

AbstractTwo different generalizations of the Perron—Frobenius theory to the matrix pencil Ax = λBx are discussed, and their relationships are studied. In one generalization, which was motivated by economics, the main assumption is that (B − A)−1A is nonnegative. In the second generalization, the main assumption is that there exists a matrix X ⩾ 0 such that A = BX. The equivalence of these two assumptions when B is nonsingular is considered. For ρ(|B−1A|) < 1, a complete characterization, involving a condition on the di-graph of B−1A, is proved. It is conjectured that the characterization holds for ρ(B−1A) < 1, and partial results are given for this case