{M}$, the column rank of ${\mathcal{O}}_{{\Upsilon}}$ is ${M}$, which results in an observable system.

To extend this proof to matrices $A = {P}{\Lambda}{P}^{-1}$, note that $${\mathcal{O}}_{{\Upsilon}} = 
 \begin{bmatrix}
  {\ensuremath{K}}A^{t_1}\\
  \cdots\\
  {\ensuremath{K}}A^{t_{L}}
 \end{bmatrix}
 =
  \begin{bmatrix}
  {\ensuremath{K}}{P}{\Lambda}^{t_1}{P}^{-1}\\
  \cdots\\
  {\ensuremath{K}}{P}{\Lambda}^{t_{L}}{P}^{-1}.
 \end{bmatrix}
 =
 \begin{bmatrix}
  {\ensuremath{K}}& \cdots & {\ensuremath{K}}\end{bmatrix}
 \boldsymbol{{P}}
 \boldsymbol{{\Lambda}}^t
 \boldsymbol{{P}^{-1}},$$ where $\boldsymbol{{P}}\in{\mathbb{R}}^{{M}{L}\times{M}{L}}$, $\boldsymbol{{\Lambda}}^t\in{\mathbb{R}}^{{M}{L}\times{M}{L}}$, and $\boldsymbol{{P}^{-1}}\in{\mathbb{R}}^{{M}{L}\times{M}{L}}$ are the block diagonal matrices associated with the system. Since $\boldsymbol{{P}}$ is an invertible matrix, the conclusions about the column rank drawn before still hold, and the system is observable.

When the eigenvalues of the system matrix are repeated, it is not enough for ${\ensuremath{K}}$ to be shaded. The next proposition proves a lower bound on the number of observations required.

\[prop:2\] Suppose that the conditions in Proposition \[prop:1\] hold, with the relaxation that the Jordan blocks $\begin{bmatrix}{\Lambda}_1 & {\Lambda}_2 &\cdots & {\Lambda}_{{O}}\end{bmatrix}$ may have repeated eigenvalues. Let ${r}$ be the number of unique eigenvalues of $A$, and let ${\gamma}({{\lambda}}_i)$ denote the geometric multiplicity of eigenvalue ${{\lambda}}_i$. Then there exist kernels ${k}(x,y)$ such that the lower bound ${l}$ on the number of sampling locations ${N}$ is given by the cyclic index of $A$, which can be computed as $$\begin{aligned}
{\label{eq:geom_mult}}
  {l}= \max_{1\leq i\leq{r}}{\gamma}({{\lambda}}_i).
 \end{aligned}$$

We first prove the 