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arXiv · 2507.10327

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

Nathaniel Johnston·Sarah Plosker·Charles Torrance·Luis M. B. Varona
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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.

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Nathaniel Johnston, Sarah Plosker, Charles Torrance, Luis M. B. Varona. 2025-07-14. Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products. https://arxiv.org/abs/2507.10327

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