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

TL;DR

This paper establishes bounds on the row sum arithmetic mean of a non-negative matrix using Perron roots of row-permuted matrices, providing conditions for equality and algorithms for positive matrices.

Contribution

It introduces bounds on the mean row sum via spectral radii of permuted matrices and offers criteria and algorithms for identifying matrices achieving these bounds.

Findings

The mean row sum is bounded by the spectral radii of row-permuted matrices.

Necessary and sufficient conditions for equality in the bounds.

Algorithms for finding matrices with extremal spectral radii when entries are positive.

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 .