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?