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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 ","authors_text":"Vanni Noferini","cross_cats":[],"headline":"The Jordan canonical form of the Fréchet derivative of f(A) is determined by the Jordan form of A and the polynomial f.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.RA","submitted_at":"2025-12-09T09:31:22Z","title":"The Jordan canonical form of the Fr\\'{e}chet derivative of a matrix function and the bivariate Jordan problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.08399","kind":"arxiv","version":5},"verdict":{"created_at":"2026-05-16T23:42:14.238464Z","id":"e45d258d-01cf-4bb4-950b-83464eecc913","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"e45d258d-01cf-4bb4-950b-83464eecc913"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a871ef45eddd9c26766a70b52fb9f9f992c38f19d90accae0caba057fa6b484e","target":"record","created_at":"2026-06-19T16:12:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5e601413c59c62c630b43fdc5dbfb81c0432fdf41313e9fda11a7d4411254d90","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.RA","submitted_at":"2025-12-09T09:31:22Z","title_canon_sha256":"2db9201129b2fe874ef45ae8946ffa18a2986919a7fba34f7bf5973083c9bb8c"},"schema_version":"1.0","source":{"id":"2512.08399","kind":"arxiv","version":5}},"canonical_sha256":"af92280625f8d4dde0e2c97c5022034ba5d596e41ac2f82a48eed33fa18fb620","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af92280625f8d4dde0e2c97c5022034ba5d596e41ac2f82a48eed33fa18fb620","first_computed_at":"2026-06-19T16:12:49.694194Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:49.694194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ju1uWSwkbPp0weIGt2PuRIKs11FULoggsrdS9p9aP6bL4ulxvPYw23bCpTky5ADw6qOWMNQQbNueSoOZK/GLDw==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:49.694614Z","signed_message":"canonical_sha256_bytes"},"source_id":"2512.08399","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a871ef45eddd9c26766a70b52fb9f9f992c38f19d90accae0caba057fa6b484e","sha256:40e23e90fc09bca312e7bdf62d91255d47d2b6c78b04a5dbb41f2ec09def0e6d"],"state_sha256":"31d91f22f82f265b8f026ceac51158b4cac88c7a80b6917466b9319dc0964fe6"}