Spectral radius of Hadamard product versus conventional product for non-negative matrices

TL;DR

This paper establishes a new inequality relating the spectral radius of the Hadamard product of non-negative matrices to their conventional product, confirming a conjecture and providing bounds useful in matrix analysis.

Contribution

The paper proves a conjectured inequality for the spectral radius of Hadamard versus conventional matrix products in non-negative matrices.

Findings

Proves the inequality $ ho(A ext{ extperiodcentered} B) \,\le\, \rho((A\text{ extperiodcentered} A)(B\text{ extperiodcentered} B))^{1/2} \,\le\, \rho(AB)$.

Confirms a conjecture by X. Zhan on spectral radius bounds.

Provides new insights into spectral properties of Hadamard and conventional matrix products.

Abstract

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all $n\times n$ non-negative matrices $A$ and $B$, $\rho(A\circ B)\le\rho((A\circ A)(B\circ B))^{1/2}\le\rho(AB)$, where $\circ$ represents the Hadamard product.