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\begin{enumerate}  \item $D$ and $\hat D$ are block diagonal and positive definite matrices; \item both $\mC$ and $\hat \mC$ are stable and minimum phase,  \end{enumerate}  then there exists a paraunitary matrix $\mathcal V(z)$ such that \cite{anderson3} \cite{Anderson_1982}  \begin{align}  \hat \mC(z)=\mC(z)\mathcal V(z),\label{eq:c1c2}\\  \hat D= \mathcal V(z) D\mathcal V^*(z).\label{eq:q1q2}  \end{align}  From \eqref{eq:c1c2}, since both $\mC(z)$ and $\hat \mC(z)$ are stable and minimum phase, $\mathcal V(z)$ is stable and minimum phase, which implies that $\mathcal V(z)$ is a constant matrix independent of $z$ \cite{anderson, hayden}. \cite{Anderson_1969, Hayden_2014}.  Therefore, we denote it simply as $V$. Take $z\rightarrow\infty$ on both sides of \eqref{eq:c1c2} yields \begin{equation}  \begin{bmatrix}  0 & I \\