How come in the following equation: $$\sum_{r=0}^\infty\frac{rt^{r-1}A^r}{r!}=\sum_{r=0}^\infty\frac{(tA)^{r-1}A}{(r-1)!}$$ when you divide by $r$ from the LHS, the power of r on matrix A goes away in the series on the RHS. I understand when you divide by r the factorial goes from $r!$ to $(r-1)!$ What property allows you to divide by $r$ from $A^r$ to get $A$?
Thanks, I'm trying to be a physics major but my math background isn't the best. This website has been helping!
Actually, the power of $r$ does not go away. It has been disguised: $t^{r-1}A^r=t^{r-1}A^{r-1}A=(tA)^{r-1}A$, the last equality holding because scalars are freely swappable in a matrix product.