Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Mallik, Debanjan
Institution: University of Maryland, College Park
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211907779