# Condition for controllability of matrix pairs

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!!

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maybe you can use Kalman's rank theorem which says system is controllable iff rank of a control llable matrix is full rank. –  Detectives Dec 2 '12 at 18:07
There is a theorem which says that (A,b) is controllable iff [b, Ab, A^2b, a^(N-1)b] has full row rank. But in this case, what is the structure of [b, Ab, ..., A^(n-1)b] ? –  Deeya Dec 2 '12 at 22:02