PBH Test matrix

Question

Can the matrix occurring from PBH test have rank greater than the dimension n of the state space? What does it mean for the controllability of the system?

Answer 1

No. As mentioned in the comments, the PBH matrix is defined to be

$$ K = [(A-\lambda I) B].$$

Since $A$ is $n\times n$ and $B$ is $n \times m$, it follows that $K$ is $n \times (n+m) $. Since the rank of a matrix cannot exceed the lesser of the column or row numbers, it follows that the rank of $K$ is at most $n$.