Pith Number

pith:V6JCQBRF

pith:2025:V6JCQBRF7DKN3YHCZF6FAIQDJO

not attested not anchored not stored refs pending

The Jordan canonical form of the Fr\'{e}chet derivative of a matrix function and the bivariate Jordan problem

Vanni Noferini

The Jordan canonical form of the Fréchet derivative of f(A) is determined by the Jordan form of A and the polynomial f.

arxiv:2512.08399 v5 · 2025-12-09 · math.RA

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{V6JCQBRF7DKN3YHCZF6FAIQDJO}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-06-19T16:12:49.694194Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

af92280625f8d4dde0e2c97c5022034ba5d596e41ac2f82a48eed33fa18fb620

Aliases

arxiv: 2512.08399 · arxiv_version: 2512.08399v5 · doi: 10.48550/arxiv.2512.08399 · pith_short_12: V6JCQBRF7DKN · pith_short_16: V6JCQBRF7DKN3YHC · pith_short_8: V6JCQBRF

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/V6JCQBRF7DKN3YHCZF6FAIQDJO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: af92280625f8d4dde0e2c97c5022034ba5d596e41ac2f82a48eed33fa18fb620

Canonical record JSON

{
  "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
  }
}