5

Gramians

qLet *A* be stable. Then a pair(*C,A*) is observable iff the *observability Gramian Q* > 0

Similarly,
(*A,B*) is controllable iff the *controllability **Gramian P* > 0

qWhen A is not necessarily
stable, the above Lyapunov equations may still have solutions. But they
are not gramians.