Lemma (Equivalent characterization of minimal)

Also, a LTI Realization $(A,B,C,D)$ where $A\in M_{n}(\mathbb{R})$, of a given Weighting Pattern is called minimal if $\not\exists$ any Equivalent Realization $(\tilde{A},\tilde{B},\tilde{C},\tilde{D})$ where $\tilde{A}\in M_{\tilde{n}}$ with $\tilde{n}<n$.