Let's say that I have a system matrix A and to find out the eigenvalues $\lambda$ ,I do this:
$$ \hbox{det}(\lambda I - A) = 0 $$
Then to find out if the system are controllable, I uses the Hautus Lemma test . This thest is mutch better that the regular $\hbox{rank}(\hbox{ctrb}(A, B)) = n\ $ test. Anyway! Here it is:
$$ \hbox{rank}([\lambda_i I - A, B]) = n$$
Let's say that I got 3 eigenvalues of A. They are $\lambda_1 = -2$ , $\lambda_2 = -10$ and $\lambda_3 = -0.5$. Now I test if the system is controllable: \begin{align} \hbox{rank}([\lambda_1 I - A, B]) &= 3\, ,\\ \hbox{rank}([\lambda_2 I - A, B]) &= 1\, ,\\ \hbox{rank}([\lambda_3 I - A, B]) &= 3 \end{align}
So something went wrong here! I got 3 eigenvalues, which mean that my state vector is the length 3. That means that my rank of the system should be number 3. But this: $$ \hbox{rank}([\lambda_2 I - A, B]) = 1$$ gives number 1 buy using $\lambda_2 = 10$.
Question: Does this mean that something is wrong with my state vector at row number 2 beacuse the eigenvalue $\lambda_2 = 10$ must reprecent the state vector $x_2$ ?