What are controllability and observability in state-space systems?

Controllability means the system can be driven from any initial state to any final state in finite time using appropriate input. The controllability matrix Qc = [B AB A^2B ... A^(n-1)B] must be full rank. Observability means all states can be determined from output measurements over time. The observability matrix Qo = [C CA CA^2 ... CA^(n-1)]^T must be full rank. These properties are essential for controller and observer design - a state feedback controller can only place poles of controllable modes.