Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$?
Now to check controllability I usually check the rank of the matrix $$R = \left( B\ \ \ \ AB\ \ \ \ A^2B\ \ \ \ \ldots \ \ \ A^{n-1}B\right).$$ This rank has to equal $n$ where $n$ is the dimension of the $A$ matrix. Now if this matrix has to have rank $n$ for every non zero $B$ we can conclude something like $A$ has full rank or something or that $A$ spans $\Bbb R^n$ or something... I am sorry for not really having a solid idea on how to solve this... I tagged this linear algebra even though its a question from a systems control class because the solution probably needs only linear algebra.
Thanks for any help!