bibkey: fan1949weyl authors: Ky Fan year: 1949 title: On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations I doi: 10.1073/pnas.35.11.652 url: https://pmc.ncbi.nlm.nih.gov/articles/PMC1063101/ claim: Theorem 1 on page 653 gives the maximum and minimum sums of Hermitian quadratic forms over orthonormal vectors as the corresponding extreme eigenvalue sums. strata_touched: [] license: citation-only triage: anchor

Spectral extremum for a declared response loss

Proceedings of the National Academy of Sciences 35(11), 652–655. Crossref, the PMC bibliographic record and the original page scans agree on the title, author and DOI. Theorem 1 on scanned page 653 states that for a Hermitian transformation with decreasing eigenvalues, the largest and smallest sums of q eigenvalues are respectively the maximum and minimum of the sum of its quadratic forms on q orthonormal vectors. The proof on that page expands in an orthonormal eigenbasis.

The FIB boundary geometry volume §23 uses the maximum for a finite response Gram matrix. Its factor of three comes from the declared pure vector representation, its expected squared loss from the stated identity covariance, and its maintenance threshold from a specified resource function. Finite-horizon spectral approximation does not prove invariance under repeated projected dynamics or sufficiency for arbitrary new contexts. This note does not attribute those additional contracts to Fan’s theorem.