bibkey: cadavidrodriguezvidal2023nilpotency
authors: Paula Cadavid; Pablo M. Rodriguez; Sebastian J. Vidal
year: 2023
title: "Hilbert evolution algebras, weighted digraphs, and nilpotency"
doi: 10.48550/arXiv.2311.08387
url: https://arxiv.org/abs/2311.08387v2
claim: "Examples 3.6-3.7 rotate equal-square pairs to obtain different complete natural orthonormal bases."
strata_touched:
  - D5/S3/QuadraticForms/EvolutionAlgebras/HilbertNaturalBasisNonuniqueness
license: citation-only
triage: anchor

Equal-square orthonormal basis changes

Examples 3.6-3.7 exhibit complete natural orthonormal basis changes by rotating equal-square pairs and normalizing. Equality of two squares makes the mixed product cancel under a real orthogonal block. This mechanism applies over complex scalars as well. The displayed examples themselves have all square vectors in a proper closed coordinate hyperplane; their algebraic squares are not norm dense in the whole Hilbert space. The rotation method is a supporting ingredient and is distinct from a dense-square counterexample.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2311.08387
• URL: https://arxiv.org/abs/2311.08387v2
• Text: https://arxiv.org/html/2311.08387v2
• Locations: section 3.2, Examples 3.6-3.7; section 4 for the algebraic square convention.