sufficient for the system to be observable. In Proposition \[prop:2\], we prove a lower bound on the number of sampling locations required for observability which holds for more general $A$. Finally, in Proposition \[prop:3\], we outline a method that achieves this lower bound for certain kernels. Since both ${\ensuremath{K}}$ and ${{\ensuremath{K}}_{{C}{D}}}$ are kernel matrices generated from a shared kernel, these observability results translate directly into controllability results.

To prove our results, we will leverage the spectral decomposition of $A$. Specifically, recall that any matrix $A\in{\mathbb{R}}^{{M}\times{M}}$ is similar to a unique block diagonal matrix ${\Lambda}$ (i.e. $\exists P\in{\mathbb{R}}^{{M}\times{M}}$ invertible such that $A = P{\Lambda}P^{-1}$) whose diagonal blocks are matrices of the form $$\begin{aligned}
{\Lambda}_k(\lambda_i, \lambda_i^*) := 
\begin{bmatrix}
{M}& {I_2}& \cdots &  {0}\\
\vdots & \vdots & \ddots  & {I_2}\\
{0}& {0}& \cdots & {M}{\label{eq:jor_com}}
\end{bmatrix}.\end{aligned}$$ where $(\lambda_i, \lambda_i^*)$ is a complex conjugate eigenvalue of $A$, and $
{M}= \begin{bmatrix}
\mu_1 & \mu_2\\
-\mu_2 & \mu_1
\end{bmatrix}$ and $ {I_2}= \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}$. Real eigenvalues ${\lambda}_i$ correspond to the case ${M}= {\lambda}_i$ and ${I_2}= 1$. Thus the complete real Jordan form of $A$ will be the appropriate diagonal array of these blocks. If all the eigenvalues $\lambda_i$ 