# a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix $\exp(t(U+sH))$, can someone tell me what is the derivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers).

Thus,if I let $A(s,t)=\exp(t(U+sH))$,and suppose$$B(s,t)=(A(s,t))^{-1}{{\partial A \over \partial s}{(s,t)}},$$ then prove that $B(s,t)$ is infinitely differentiable .

I know that ${\partial A\over \partial t}(s,t)=exp(t(U+sH))(U+sH)$,but I am not sure how to prove B(s,t) is infinitely differentiable. Any help?

• actually, I have no idea how to write ${\partial A\over \partial s}(s,t)$ down,I mean is there some expression to represent ${\partial A\over \partial s}(s,t)$ – python3 Apr 24 '15 at 2:54
• Calculus for matrix functions can be found in, e.g., Bhatia's Matrix Analysis, Theorem V.3.3. – gerw Apr 24 '15 at 5:57
• There's no clean expression unless $U$ and $V$ commute. You can define the exponential with a power series and differentiate term-by-term $\sum_{n=0}^{\infty}\frac{t^{n}}{n!}(U+sV)^{n}$. – DisintegratingByParts Apr 28 '15 at 1:20