each other. Then system [(\[eq:k\_measure\])]{} is observable if ${\Upsilon}$ has distinct values, and $|{\Upsilon}| \geq {M}$.

A cyclic index of ${l}$ for this system implies that there exists an eigenvalue ${{\lambda}}$ that’s repeated ${l}$ times. WLOG, let ${\mathbf{{\ensuremath{K}}}}$ have ${l}$ fully shaded, linearly independent rows, and, assume that the column indices corresponding to this eigenvalue are $\{1,2,\dots,{l}\}$. Define ${\boldsymbol{{{\lambda}}}}_i:= \begin{bmatrix}{{\lambda}}_i^{t_1} & {{\lambda}}_i^{t_2} & \cdots {{\lambda}}_i^{t_{{L}}}\end{bmatrix}^T$. Then $$\begin{aligned}
  {\mathcal{O}}_{{\Upsilon}}
   &:= 
   \begin{bmatrix}
    k_{11} {\boldsymbol{{{\lambda}}}}_1 & k_{12} {\boldsymbol{{{\lambda}}}}_2 & \cdots & k_{1{M}} {\boldsymbol{{{\lambda}}}}_{{M}}\\
    \vdots & \vdots & \ddots & \vdots\\
    k_{{l}1} {\boldsymbol{{{\lambda}}}}_1 & k_{{l}2} {\boldsymbol{{{\lambda}}}}_2 & \cdots & k_{{l}{M}} {\boldsymbol{{{\lambda}}}}_{{M}}.
   \end{bmatrix}
 \end{aligned}$$ Let ${\boldsymbol{{{\lambda}}}}_1 = {\boldsymbol{{{\lambda}}}}_2 = \cdots {\boldsymbol{{{\lambda}}}}_{{l}} := {\boldsymbol{{{\lambda}}}}$. Focusing on these first ${l}$ columns of this matrix, this implies that we need to find constants $c_1,c_2,\dots, c_{{l}-1}$ s.t. $$\begin{aligned}
  \begin{bmatrix}
   k_{11}\\
   \vdots\\
   k_{{l}1}
  \end{bmatrix}
  &= 
  c_1 
  \begin{bmatrix}
   k_{12}\\
   \vdots\\
   k_{{l}2}
  \end{bmatrix}
  + \cdots + 
  c_{{l}-1} 
  \begin{bmatrix}
   k_{1{l}}\\   
   \vdots\\
   k_{{l}{l}}. 
  \end{bmatrix}
 \end{aligned}$$ However, these columns are linearly independent by assumption, and thus no such constants exist, implying that ${\mathcal{O}}_{{\Upsilon}}$ is observable.

An example of a kernel such that any subset of ${l}$ columns in ${\mathbf{{\ensuremath{K}}}}$ are linearly independent of each other is the Gaussian kernel evaluated on sampling locations ${\{x_1, \dots, 