Let $X,Y$ be two $n\times n$ complex matrices. Consider the function $f(t)=e^{tX}e^{tY}$. Is it correct that $\frac{d}{dt}f(t)=Xe^{tX}e^{tY}+e^{tX}Ye^{tY}$?
Otherwise, how to prove that $X +Y$ is the tangent vector to $f(t) = e^{tX} e^{tY}$ at $1$?
2026-04-09 16:33:12.1775752392
Product rule of exponential matrix differentiation
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