full column rank of observability matrix if $C \succ 0$

Question

Suppose I know that $C\succ 0$, and that $A$ is asymptotically stable.

Can I say the observability matrix $\mathcal{O}$ is full column rank?

where $\mathcal{O} = \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1}\end{bmatrix}$

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$A$ is asymptotically stable implies all eigenvalues of $A$ are of negative real part.

Answer 1

Since $C\succ0$ implies that $C$ is already full rank, then the observability matrix $\mathcal{O}$ is also always full rank, independently of $A$. Because the rank of $\mathcal{O}$ should at least be equal to that of $C$.