An analog of H\"older's inequality for the spectral radius of Hadamard products

TL;DR

This paper establishes new inequalities for the spectral radius of Hadamard products of matrices, extending H"older's inequality to this context and providing sharper bounds for specific cases.

Contribution

The paper introduces an analog of H"older's inequality for the spectral radius of Hadamard products and derives sharper bounds for particular parameter choices.

Findings

Proved a H"older-type inequality for spectral radius of Hadamard products.

Derived a sharper inequality for the case p=q=2.

Analyzed special case p=1, q=∞ for spectral radius bounds.

Abstract

We prove new inequalities related to the spectral radius $\rho$ of Hadamard products (denoted by $\circ$) of complex matrices. Let $p,q\in [1,\infty]$ satisfy $\frac{1}{p}+\frac{1}{q}=1$, we show an analog of H\"older's inequality on the space of $n\times n$ complex matrices $$\rho(A\circ B) \le \rho(|A|^{\circ p})^{\frac{1}{p}} \rho(|B|^{\circ q})^{\frac{1}{q}} \quad \text{for all $A,B\in \mathbb{C}^{n\times n}$,} $$ where $|\cdot|$ denotes entry-wise absolute values, and $(\cdot)^{\circ p}$ represents the entry-wise Hadamard power. We derive a sharper inequality for the special case $p=q=2$. Given $A,B\in \mathbb{C}^{n\times n}$, for some $\beta \in (0,1]$ depending on $A$ and $B$, $$\rho(A\circ B) \le \beta \rho(|A\circ A|)^{\frac{1}{2}} \rho(|B\circ B|)^{\frac{1}{2}} .$$ Analysis for another special case $p=1$ and $q=\infty$ is also included.