This is a cross-post to [the question](https://math.stackexchange.com/questions/2952997/is-there-a-homeomorphism-between-the-sets-of-schur-stable-and-hurwitz-stable-mat) I asked at MSE.




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The set of Schur stable matrices is
\begin{align\*}
\mathcal S = \{A \in M\_n(\mathbb R): \rho(A) < 1\},
\end{align\*}
where $\rho(\cdot)$ denotes the spectral radius of a matrix and the set of Hurwitz stable matrices is 
\begin{align\*}
\mathcal H = \{A \in M\_n(\mathbb R): \max\_{i=1, \dots, n} \text{Re}(\lambda\_i(A)) < 0\},
\end{align\*}
i.e., matrices with eigenvalues lying on the open left half plane.
Let $\mathcal C$ denote all matrices in [companion form](https://en.wikipedia.org/wiki/Companion_matrix). Let $\hat{\mathcal S} = \mathcal S \cap \mathcal C$ and $\hat{\mathcal H} = \mathcal H \cap \mathcal C$. Now I would like to determine whether there is a homeomorphism between the sets $\hat {\mathcal S}$ and $\hat{\mathcal H}$. Let us exclude the trivial case $n=1$.


If we consider the sets $\mathcal S$ and $\mathcal H$ only, there is a diffeomorphism $f: \mathcal S \to \mathcal H$ given by
\begin{align\*}
A \mapsto (A-I)^{-1}(A+I).
\end{align\*}
But apparently this doesn't work for $\hat {\mathcal S}$ and $\hat{\mathcal H}$ since the inversion and multiplication will not necessarily yield a matrix in companion form. 



Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?
Essentially you're asking if there is a homeomorphism between the monic real polynomials of degree $n$ with roots in the open unit disk and those with roots in the left half plane. Just take $$\prod\_{j=1}^n (x - \alpha\_j) \mapsto \prod\_{j=1}^n \left(x - \frac{\alpha\_j+1}{\alpha\_j - 1}\right)$$
This should work because the multiset of roots of a polynomial is a continuous function of the coefficients.


