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

Abstract

$R_+^{n\times n}$ denotes the set of $n\times n$ non-negative matrices. For $A\in R_+^{n\times n}$ let $\Omega(A)$ be the set of all matrices that can be formed by permuting the elements within each row of $A$. Formally: $$\Omega(A)=\{B\in R_+^{n\times n}: \forall i\;\exists\text{ a permutation }\phi_i\; \text{s.t.}\ b_{i,j}=a_{i,\phi_i(j)}\;\forall j\}.$$ For $B\in\Omega(A)$ 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 $\Omega(A)$ Formally, we are showing that $$\min_{B\in\Omega(A)}\rho(B)\leq \frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n a_{i,j}\leq \max_{B\in\Omega(A)}\rho(B).$$ For positive $A$ we also obtain necessary and sufficient conditions for one of these inequalities (or, equivalently, both of them) to become an equality. We also give criteria which an irreducible matrix $C$ should satisfy to have $\rho(C)=\min_{B\in\Omega(A)} \rho(B)$ or $\rho(C)=\max_{B\in\Omega(A)} \rho(B)$. These criteria are used to derive algorithms for finding such $C$ when all the entries of $A$ are positive .