Inequalities for the Perron root related to Levingerʹs theorem Original Research Article

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 varrho (A + D−1BTD) greater-or-equal, slanted varrho(A) + varrho(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 varrho(αA + (1 − α)BT) greater-or-equal, slanted αvarrho(A) + (1 − α)varrho(B), proved under stronger assumptions by Bapat, and implies a generalization of Levingerʹs 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 varrho(A(α) c(D−1ATD)(c−α)), c greater-or-equal, slanted 1, 0 less-than-or-equals, slant α less-than-or-equals, slant c, of a weighted (entrywise) geometric mean of A and D−1ATD, where A(α) = (αijα) and “o” denotes the Hadamard product, the monotonicity property dual to that asserted by generalized Levingerʹs theorem is established.