{"paper":{"title":"The Jordan canonical form of the Fr\\'{e}chet derivative of a matrix function and the bivariate Jordan problem","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"The Jordan canonical form of the Fréchet derivative of f(A) is determined by the Jordan form of A and the polynomial f.","cross_cats":[],"primary_cat":"math.RA","authors_text":"Vanni Noferini","submitted_at":"2025-12-09T09:31:22Z","abstract_excerpt":"Let $\\mathbb{F}$ be an algebraically closed field of characteristic $0$. Given a square matrix $A \\in \\mathbb{F}^{n \\times n}$ and a polynomial $f \\in \\mathbb{F}[w]$, we determine the Jordan canonical form of the formal Fr\\'{e}chet derivative of $f(A)$, in terms of that of $A$ and of $f$. When $\\mathbb{F}\\subseteq \\mathbb{C}$, via Hermite interpolation, our result provides a solution to [N.J. Higham, \\emph{Functions of Matrices: Theory and Computation}, Research Problem 3.11]. A generalization consists of finding the Jordan canonical form of linear combinations of Kronecker products of powers "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Given a square matrix A in F^{n x n} and a polynomial f in F[w], we determine the Jordan canonical form of the formal Fréchet derivative of f(A), in terms of that of A and of f.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The field F is algebraically closed of characteristic zero and f is a polynomial; the bivariate generalization requires further assumptions for the partial results to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Jordan canonical form of the Fréchet derivative of f(A) for polynomial f is explicitly determined from the Jordan form of A and f.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Jordan canonical form of the Fréchet derivative of f(A) is determined by the Jordan form of A and the polynomial f.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"66821f96661047201a865e32cb888d707c5e4640f1fa8c04aba9d3a7f4ab333d"},"source":{"id":"2512.08399","kind":"arxiv","version":5},"verdict":{"id":"e45d258d-01cf-4bb4-950b-83464eecc913","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T23:42:14.238464Z","strongest_claim":"Given a square matrix A in F^{n x n} and a polynomial f in F[w], we determine the Jordan canonical form of the formal Fréchet derivative of f(A), in terms of that of A and of f.","one_line_summary":"The Jordan canonical form of the Fréchet derivative of f(A) for polynomial f is explicitly determined from the Jordan form of A and f.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The field F is algebraically closed of characteristic zero and f is a polynomial; the bivariate generalization requires further assumptions for the partial results to hold.","pith_extraction_headline":"The Jordan canonical form of the Fréchet derivative of f(A) is determined by the Jordan form of A and the polynomial f."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.08399/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"f2a3c859e123251e2ef09a5fcbd43b85453f3ab735223f4d9bc3c252565112f1"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}