Admissible, Stabilizability, and Bicoprime Factorization
In [1], a system $G=\begin{bmatrix} G_{11} & G_{12}\\ G_{21} & G_{22}\end{bmatrix}$ is admissible if the characteristic determinant (i.e., determinant of the denominator) of a coprime factorization of $G$ is equivalent to the characteristic determinant of a coprime factorization of $G_{22}$. It says that admissibility plays the same roles as joint stabilizability/detectability plays in the state-space theory.
Let $G = NM^{-1}=\tilde{M}^{-1}\tilde{N}$ is the right and left coprime factorization of $G$, respectively. Then [2] shows that $G$ is stabilizable is equivalent to that
$\left(M,~\begin{bmatrix} 0 & I\end{bmatrix}N\right)$ is right-coprime and $\left(M,~\begin{bmatrix}\begin{smallmatrix} 0 \\ I \end{smallmatrix}\end{bmatrix}\right)$ is left-coprime,
or
$\left(\tilde{M},~\tilde{N}\begin{bmatrix}\begin{smallmatrix} 0 \\ I\end{smallmatrix}\end{bmatrix}\right)$ is left-coprime and $\left(\tilde{M},~\begin{bmatrix} 0 & I \end{bmatrix}\right)$ is right-coprime.
Note that if the coprimeness holds, both $\left(\begin{bmatrix} 0 & I\end{bmatrix}NM^{-1}\begin{bmatrix}\begin{smallmatrix} 0 \\ I\end{smallmatrix}\end{bmatrix}\right)$ and $\left(\begin{bmatrix} 0 & I \end{bmatrix}\tilde{M}^{-1}\tilde{N}\begin{bmatrix}\begin{smallmatrix} 0 \\ I\end{smallmatrix}\end{bmatrix}\right)$ are actually bicoprime factorizations (right-left coprime factorizations) of $G_{22}$. From [3], the characteristic determinant of $G_{22}$ is $\det M$, this means that the admissibility of [1] is equivalent to the stabilizability of [2], as the characteristic determinant of $G$ is also $\det M$.
Example. Suppose $G = \begin{bmatrix} G_{11} & G_{12}\\ G_{21} & G_{22}\end{bmatrix}$ and $G_{11} = G_{12}= G_{21} = G_{22}$, then $G$ is stabilizable.
Let $G_{22} = NM^{-1}$ be a right coprime factorization of $G_{22}$. Then it is easy to show that
\begin{align*}
\begin{bmatrix} I \\ I \end{bmatrix} N \begin{bmatrix} I & I \end{bmatrix} \begin{bmatrix} M & M-I \\ 0 & I \end{bmatrix}^{-1}
\end{align*}
is a right coprime factorization of $G$. The rest is to show that this factorization indeed satisfies the left and right coprimeness conditions above. Thus, $G$ is stabilizable/admissible.
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