Let $A\in M_{n}(\Bbb R)$ and $B\in M_{n,m}(\Bbb R)$ and $C=\int_{0}^{1}\exp\left(sA\right)BB^T\exp\left(sA^T\right)\,{\rm d}s$.
Prove that $C$ is invertible if and only if $\sum_{i=0}^{n-1} Im(A^iB)=\Bbb R^n$
I found this weird-looking challenging problem in "Revue de la filière mathématique". I have tried to solve it in specific cases such as $m=1$ or $n=1$ but can't find the way for the general case. It's really mysterious and surprising to me how the integral, the exponential function and the properties of linear operators can have a beautiful relationship between them.
I've get stuck on the problem for days and I need some hints :) (Here we regard a matrix $M\in M_{n,m}(\Bbb R)$ as a linear transformation from $\Bbb R^m$ to $\Bbb R^n$)