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PreprintarXiv licencearXiv

Hadamard powers of Gram matrices and Generalized Cauchy-Schwarz inequalities

  • Nathaniel Johnston
  • Mohammad Sababheh
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Abstract

The following inequality for vectors v, w ∈ ℝn was considered in [Linear and Multilinear Algebra 74:1779--1798 (2026)]: |vp||wp| − ⟨vp,wp⟩ ≤ |v|p|w|p − ⟨v,w⟩p, where vp is the vector that is obtained by raising each entry of v to the power p. It was shown that this inequality holds for all positive integers p, and it was conjectured that it holds for all real p ≥ 2 when the entries of v and w are positive. A formula for the maximum amount by which the inequality's right-hand side can exceed its left-hand side for nonnegative unit vectors was also conjectured. We prove both of these conjectures, and we determine exactly when the inequality holds with equality. We also present a new generalization of this inequality to three or more vectors.

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Cite

BibTeXRIS
BibTeX
@misc{johnston2026hadamard,
  title = {{Hadamard powers of Gram matrices and Generalized Cauchy-Schwarz inequalities}},
  author = {Nathaniel Johnston and Mohammad Sababheh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12063},
  eprint = {2610.12063},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12063},
}
RIS
TY  - UNPB
TI  - Hadamard powers of Gram matrices and Generalized Cauchy-Schwarz inequalities
AU  - Nathaniel Johnston
AU  - Mohammad Sababheh
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12063
UR  - https://arxiv.org/abs/2610.12063
AB  - The following inequality for vectors v, w ∈ ℝ^n was considered in [Linear and Multilinear Algebra 74:1779--1798 (2026)]: |v^p||w^p| − ⟨v^p,w^p⟩ ≤ |v|^p|w|^p − ⟨v,w⟩^p, where v^p is the vector that is obtained by raising each entry of v to the power p. It was shown that this inequality holds for all positive integers p, and it was conjectured that it holds for all real p ≥ 2 when the entries of v and w are positive. A formula for the maximum amount by which the inequality's right-hand side can exceed its left-hand side for nonnegative unit vectors was also conjectured. We prove both of these conjectures, and we determine exactly when the inequality holds with equality. We also present a new generalization of this inequality to three or more vectors.
ER  -