Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable.
Recall that such a $T$ is said to have a right pole of order $r$ at direction $v$ at the point $z_0$ if $v(z)$ is a vector valued function holomorphic near $z_0$, such that $v(z_0) \ne 0$ and such that the vector valued function $T(z)v(z)$ has a pole of order $r$ at $z_0$.
It is well known that if such a $T$ is holomorphic at infinity, then it may be represented by an equation of the form
$T(z) = D+C(zI-A)^{-1}B$
Furthermore, assuming $A$ is of size $n$ and $T$ has $n$ poles (i.e, $T$ has a maximal McMillan degree - a representation with $n$ small as possible) then this representation is unique up to similarity: any other matrices $A',B',C',D'$ which represents $T$ are of the form $D'=D$, $A' = N^{-1}AN$, $B'=N^{-1}B$, $C'=CN$, where $N$ is some invertible matrix
My question: Suppose we are given two meromorphic matrix functions $T_1(z) = D+C(zI-A)^{-1}B$ and $T_2 = D'+C'(zI-A')^{-1}B'$. Suppose both of these are of maximal McMillan degree, $A$ and $A'$ are of the same size, and supopse they share the same pole data.
Does it follows that there exists an invertible matrix $N$ such that $A'=N^{-1}AN$ and $B'=N^{-1}B$?
Thanks!