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On the differential of the exponential map
Abstract
We study the time derivative of the matrix exponential B(t)=exp(A(t)), where A(t) is a time-parametrized curve in 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.
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BibTeX
@misc{mallik2026differential,
title = {{On the differential of the exponential map}},
author = {Debanjan Mallik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04762},
eprint = {2609.04762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04762},
}
RIS
TY - UNPB
TI - On the differential of the exponential map
AU - Debanjan Mallik
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04762
UR - https://arxiv.org/abs/2609.04762
AB - We study the time derivative of the matrix exponential B(t)=exp(A(t)), where A(t) is a time-parametrized curve in 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.
ER -