Is the pair controllable/observable?

Question

The matrices $Q\in\mathbb R^{n\times n}$ and $G\in\mathbb R^{n\times n}$ are both symmetric positive semidefinite, $A\in\mathbb R^{n\times n}$ is invertible. Moreover, $(A,G)$ is controllable, and $(Q,A)$ is observable. I have the following questions

• Is $(A,-G)$ controllable?
• Is $(-Q,A)$ observable?

Thanks in advance!

Answer 1

Short answer: yes and yes.

Actually for an $n \times m$ matrix $G$ it is true. Just write the controllability matrix to see that. The same thing goes for observability.