I need to find geometric interpretation of PBH test i.e. for any space X isomorphic to R^n and U isomorphic to R^m. A is a linear operator from X to X and B is a linear operator from U to X. Prove that the pair (A,B) is controllable if and only if X=im(A-lambda*I)+im B. where I is an identity operator from X to X and lambda is the eigen value.
I am unable to figure out how to proceed. Please help me out with this. Thank you.
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This is a problem on a midterm exam that was assigned Tuesday in a graduate level controls course. As such, please refrain from providing the answer until tomorrow, as it is due at 11:30 p.m. EST today, 4/9. It is not a coincidence that it was asked yesterday, as the exam is a take-home exam and was assigned Tuesday.
So you hopefully don't delete this post immediately, I will add the answer the the question in full tomorrow morning.
Maybe what you want is the eigenvector test: The pair $(A,B)$ is controllable iff there is no eigenvector of $A^T$ in the kernel of $B^T$.