Bounding the row sum arithmetic mean by Perron roots of row-permuted matrices

Rnxn + denotes the set of n x n non-negative matrices. For A E Rnxn + let S2(A) be the set of all matrices that can be formed by permuting the elements within each row of A. Formally: S2(A) = {B E Rnxn + : Vi a permutation & phi;i s.t. bi,j = ai,& phi;i(j) Vj}. For B E S2(A) let & rho;(B) denote the spectral radius or largest non-negative eigenvalue of B. We show that the arithmetic mean of the row sums of A is bounded by the maximum and minimum spectral radius of the matrices in S2(A). Formally, we show that ⠂n n min & rho;(B) < 1 ⠂ai,j < max & rho;(B). B & ISIN;& omega;(A) B & ISIN;& omega;(A) i j n =1 =1 For positive A we obtain necessary and sufficient conditions for these inequalities to become an equality. We also give criteria which an irreducible matrix C should satisfy so that & rho;(C) = minB & ISIN;& omega;(A) & rho;(B) or & rho;(C) = maxB & ISIN;& omega;(A) & rho;(B). Thesecriteria are used to derive algorithms for finding such C when all the entries of A are positive.& COPY; 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).