bibkey: willems1976symmetry authors: Jan C. Willems year: 1976 title: Realization of systems with internal passivity and symmetry constraints doi: 10.1016/0016-0032(76)90081-8 claim: Minimal internally symmetric realizations are related by changes of basis preserving the realization’s invariant quadratic form; the positive-metric case gives orthogonal equivalence. strata_touched:

• D5/S3/Quantum/Measurements/MinimalSymmetricRealizationUniqueness license: citation-only triage: anchor

Minimal Symmetric Realizations

Willems, Theorem II and Remark 3, describe minimal internally symmetric realizations through factorizations of a uniquely determined nonsingular symmetric form. Remark 4 identifies the associated quadratic form as independent of the chosen internally symmetric realization. The preceding proof uses uniqueness of the change of state coordinates between minimal realizations.

The Lean theorem proves the positive real inner-product specialization from equality of all moments, with the symmetry of both dynamics and reachability of both realizations explicit. It constructs the orthogonal map by transporting Gram forms on finitely supported input combinations. No passivity assumption or stability condition is imposed. This note does not claim that Willems uses the repository’s exact moment-based statement or its Lean type-class signature.

Verified locator

• DOI: https://doi.org/10.1016/0016-0032(76)90081-8
• Crossref metadata retrieved on 2026-09-06 confirmed the author, title, June 1976 publication, and DOI above.
• Author’s publication list: https://homes.esat.kuleuven.be/~jwillems/Publications.html
• Full text, pages 609-610 for the symmetry result: https://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/1976.4.pdf
• An initial download of 1976.1.pdf was a different article, Mechanisms for the Stability and Instability in Feedback Systems. It was excluded after inspecting its title; the publication list supplied the correct PDF link.