Inequalities for the Perron root related to Levinger's theorem

✍ Scribed by Yu.A. Alpin; L.Yu. Kolotilina

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 859 KB
• Volume: 283
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(98)10084-8

✦ Synopsis

For the Perron roots of square nonnegative matrices A, B, and A + D~BXD, where D is a diagonal matrix with positive diagonal entries, the inequality

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 inequalit~ p(x4 + (I -0t)B T) >i ~p(A) + (1 -~)p(9), proved under stronger assumptions by Bapat, and implies a generalization of Levinger~s theorem on the monotonicity of the Perron root of~ weighted arithmetic mean of a nonnegative matrix and its transpose. Also, for the Perron root "), l, 0.< of a weighted ~entrywise) geometric mean of A and D ~A r D, where A I~ = ~a,}/~ \ a" nd "'o'" denotes the Hadamard product, the monotonicity property dual to that asserted by generalized Levinger's theorem is established.