I have a question regarding how determine if a matrix pair is controllable.
If $A$ and $b$ are given by
A = [$\lambda_1 0 0 ...., 0 \lambda_20 0 0,...,00...\lambda_n$] (matrix) and $b = [b_1 b_1 ... b_n]^T$.
For what values of $\lambda_k$ and $b_k$, $k = 1,...,n$ is the matrix pair $(A,b)$ controllable?
Which result does one apply here and how? In addition, if one is given a pair of matrices, how does one show if or not it is controllable? Thanks!!