Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915388978102272 |
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| author | Johnston, Nathaniel Plosker, Sarah Torrance, Charles Varona, Luis M. B. |
| author_facet | Johnston, Nathaniel Plosker, Sarah Torrance, Charles Varona, Luis M. B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products Johnston, Nathaniel Plosker, Sarah Torrance, Charles Varona, Luis M. B. Functional Analysis Quantum Physics 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. |
| title | Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products |
| topic | Functional Analysis Quantum Physics |
| url | https://arxiv.org/abs/2507.10327 |