I noticed in Linear Control Theory, we may multiply matrices for controllability $\mathcal{C} \in \mathbb{R}^{n \times np}$ and observability $\mathcal{O} \in \mathbb{R}^{nq \times n}$ as follows $\mathcal{O} \mathcal{C} \in \mathbb{R}^{np \times nq}$. I do know, the independent columns of both matrices span respective controllable and observable spaces. However, I have less notion for its product. What does it represent?
Thank you!