Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

TL;DR

This paper generalizes a Cauchy-Schwarz-type inequality using Hadamard powers and tensor products, providing new proofs and extending it to matrices, different exponents, and multiple vectors.

Contribution

It introduces novel generalizations of a Cauchy-Schwarz inequality, including matrix versions, alternative exponents, and multi-vector inequalities, with improved conceptual understanding.

Findings

Three new proofs illustrating why the inequality holds

Generalization from vectors to matrices

Extension to exponents other than 2 and multiple vectors

Abstract

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq \|\mathbf{v}\|^2\|\mathbf{w}\|^2 - \langle\mathbf{v},\mathbf{w}\rangle^2$ for all $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$. We present three new proofs of this inequality that better illustrate "why" it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we derive a version of the inequality involving three or more vectors.