# Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

This is a cross-post to the question I asked at MSE.

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. 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.

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.