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Given a finite set {Ax}x∈X of nonnegative matrices, we derive joint upper and lower bounds for the row sums of the matrices D-1 A(x) D, x ∈ X, where D is a specially chosen nonsingular diagonal matrix. These bounds, depending only on the sparsity patterns of the matrices A(x) and their row sums, are used to obtain joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, bounds for the joint and lower spectral radii of a matrix set, and conditions sufficient for all convex combinations of given matrices to be Schur stable. Bibliography: 20 titles. © 2007 Springer Science+Business Media, Inc.

The paper presents a new monotonicity property of the Perron root of a nonnegative matrix. It is shown that this new property implies known monotonicity properties and also the Chistyakov two-sided bounds for the Perron root of a block-partitioned nonnegative matrix. Moreover, based on the monotonicity property suggested, the equality cases in Chistyakov's theorem are analyzed. Applications to bounding above the spectral radius of a complex matrix are presented. Bibliography: 9 titles. © 2007 Springer Science+Business Media, Inc.

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.