INEQUALITIES FOR THE PERRON ROOT RELATED TO LEVINGERS THEOREM

For the Perron roots of square nonnegative matrices A, B, and A + (D-1 BD)-D-T, where D is a diagonal matrix with positive diagonal entries, the inequality rho(A + (D-1BD)-D-T) greater than or equal to rho(A) rho(B) is proved under the assumption that A and B have a common unord ered pair of nonorthogonal right and left Perron vectors. The case of equality is analyzed. The above inequality generalizes the inequality rho(alpha A + (1 - alpha)B-T) greater than or equal to alpha rho(A) (1 - alpha)rho(B), proved under stronger assumptions by Bapat, and imp lies 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 rho(A((alpha)) circle (D(-1) A(T)D)((c-alpha))), c greater than or equal to 1, 0 less than or equal to alpha less than or equal to c, of a weighted (entrywise) geometric mean of A and D(-1)A(T)D, where A((alpha)) = (a(ij)(alpha)) and ''o'' denotes the Hadamard product, the monotonicity property dual to that asserted by generalized Levinger's theorem is established. (C) 1998 El sevier Science Inc. All rights reserved.