Lemma (Alternative characterization of equivalence)

Two Realizations $(A,B,C,D)$ and $(\tilde{A},\tilde{B},\tilde{C},\tilde{D})$ of a given Weighting Pattern are equivalent if and only if $$\forall t\geq 0:Ce^{At}B=\tilde{C}e^{\tilde{A}t}\tilde{B}\iff \forall k\in\mathbb{N}:CA^{k}B=\tilde{C}\tilde{A}^{k}\tilde{B}$$and $$D=\tilde{D}$$

Lemma (Realizations are Equivalent if Laplace Transform is Equivalent)

Two Realizations $(A,B,C,D)$ and $(\tilde{A},\tilde{B},\tilde{C},\tilde{D})$ are equivalent if and only if $$\begin{pmatrix}\begin{array}{c|c} A&B \\ \hline C &D \end{array}\end{pmatrix}(s)=\begin{pmatrix}\begin{array}{c|c} \tilde{A}&\tilde{B} \\ \hline \tilde{C} &\tilde{D} \end{array}\end{pmatrix}(s)$$$\forall s \in\mathbb{C}$ where the Transfer Functions are defined.