Let (A, B) is controllable and that rank B = n , we want to show that the geometric multiplicity of each eigenvalue of A is at most n. Any help
2026-03-27 17:50:36.1774633836
Controllable system
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Hint: Recall that the controllability of the system $(A,B)$, where $A \in \mathbb R^{n \times n}$ and $B \in \mathbb R^{n \times m}$, is equivalent to the Kalman criterion:
$$\text{rank} \left( \left[B | AB | \dots | A^{n-1}B \right] \right) = n$$