# Hw 5 ex 2

Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix $$\tilde{A}=A-BK$$ must have an eigenvalue with positive real part, to which corresponds a certain eigenvector $$v$$.
One can rewrite the Algebraic Riccati equation using $$\tilde{A}$$, where you should note the changes of signs$P\tilde{A} + \tilde{A}^T P + PBQ_u^{-1}B^T P + Qx=0$
Pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and $$v^* Q_v v$$ is zero. Which contradicts the initial assumption.