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

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.
Authors:

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

Download:

Official publication from Linear and Multilinear Algebra
Preprint from arXiv:2507.10327 [math.FA]

Cite as:

N. Johnston, S. Plosker, C. Torrance, and L. M. B. Varona. Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products. Linear and Multilinear Algebra, 74(13):1779–1798, 2026.

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Pairwise completely positive matrices and conjugate local diagonal unitary invariant quantum states – An earlier paper that inspired this one. It contains the origin of this paper’s central inequality.