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On the differential of the exponential map — arXiv:2609.04762

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.04762
License (as archived): CC0-1.0
Archived: 2026-10-16T23:12:46.350Z
SHA-256 of the archived text: 133afac1c49258c842b85a47fcc496bf167d42b63b51ab2f9f7e7a8b75bf6b64
Stable URL of this page: https://metraipolis.com/documento/dc_75601b032fc3

How to cite

On the differential of the exponential map — arXiv:2609.04762 — arXiv (en), https://arxiv.org/abs/2609.04762 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_75601b032fc3 (Metraipolis archive dc_75601b032fc3, sha256 133afac1c49258c842b85a47fcc496bf167d42b63b51ab2f9f7e7a8b75bf6b64)

BibTeX

@misc{dc75601b032fc3,
  title = {On the differential of the exponential map — arXiv:2609.04762},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.04762},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_75601b032fc3; sha256 133afac1c49258c842b85a47fcc496bf167d42b63b51ab2f9f7e7a8b75bf6b64},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.04762. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the time derivative of the matrix exponential $B(t)=\mathrm{exp}(A(t))$, where $A(t)$ is a time-parametrized curve in $\mathrm{Mat}(n)$. Starting from the Taylor series expansion, we derive a nested summation formula, which is then reformulated into a double summation. This expression is converted into an integral representation using Euler's beta function, and further expressed in terms of Lie brackets and the adjoint representation. Later, we extend this to a family of problems and verify a well-known result. Finally, we establish connections to the Gateaux and Fréchet derivatives, proving the latter's existence. These results offer a unified and explicit framework for understanding the derivative of the exponential map.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-07 in math.OC).
Categorie arXiv: Systems and Control (eess.SY) ; Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.04762 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T22:51:59.432Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-07?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).