the same dynamics matrix, as shown in Figure \[fig:kernel\_variation\]. Note also that the dynamical equations [(\[eq:k\_measure\_c\])]{} and [(\[eq:k\_measure\_c1\])]{} are *independent of the choice of domain ${\Omega}$*: different domains with different kernels may result in the same sequence of matrices ${\ensuremath{K}}_k$. This allows our results to hold for any domain over which a kernel can be defined, including examples like graphs, hidden Markov models, and strings, which are not typically studied in the controls literature, at virtually no extra complexity in implementation beyond the design of the actual sensors and actuators. This remarkable fact is why we denote our method to be *domain agnostic*.

\

Since ${\ensuremath{K}}_{k+1}$ is the kernel matrix between the data points and basis vectors, its rows are of the form ${\ensuremath{K}}_{(i)} = 
 \begin{bmatrix}
  {k}(x_i, c_1) & {k}(x_i, c_2) & \cdots & {k}(x_i, c_{{M}}) 
 \end{bmatrix}$. In systems-theoretic language, each row of the kernel matrix corresponds to a *measurement* at a particular location, and the matrix itself acts as a measurement operator. We define the *generalized observability matrix* [@zhou:bk:96] as $$\begin{aligned}
{\label{eq:obs_mat}}
 {\mathcal{O}}_{{\Upsilon}} = 
 \begin{bmatrix}
  {\ensuremath{K}}_{t_1} A^{t_1}\\  
  \cdots\\
  {\ensuremath{K}}_{t_{L}} A^{t_{L}}
 \end{bmatrix},\end{aligned}$$ where ${\Upsilon}= \{t_1, t_2, \dots, t_{{L}}\}$ are the set of instances $t_i$ when we apply the measurement operators 