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- W1975532372 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) c(D−1ATD)(c−α)), c ⩾ 1, 0 ⩽ α ⩽ 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 Levinger's theorem is established." @default.
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- W1975532372 date "1998-11-01" @default.
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- W1975532372 title "Inequalities for the Perron root related to Levinger's theorem" @default.
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- W1975532372 doi "https://doi.org/10.1016/s0024-3795(98)10084-8" @default.
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