Inequalities for the Perron root related to Levinger's theorem

Abstract

For the Perron roots of square nonnegative matrices A,B, and A + D-1BTD, where D is a diagonal matrix with positive diagonal entries, the inequality ρ(A + D-1BTD) ≥ ρ(A) + ρ(B) is proved under the assumption that A and B have a common unordered pair of nonorthogonal right and left Perron vectors. The case of equality is analyzed. The above inequality generalizes the inequality ρ(αA + (1 - α)BT) ≥ αρ(A) + (1 - α)ρ(B), proved under stronger assumptions by Bapat, and implies a generalization of Levinger's theorem on the monotonicity of the Perron root of a weighted arithmetic mean of a nonnegative matrix and its transpose. Also, for the Perron root ρ(A(α) ○ (D-1ATD)(c-α)), c ≥ 1, 0≤α≤c, of a weighted (entrywise) geometric mean of A and D-1ATD, where A(α) = (aα ij) and "○" denotes the Hadamard product, the monotonicity property dual to that asserted by generalized Levinger's theorem is established. © 1998 Elsevier Science Inc. All rights reserved.