Fdtxjr

Let us consider an affine structure $star$ of $M_4(mathbb R)$ which has following form
beginalign*
beginpmatrix
0 & * & 0 & * \
1 & * & 0 & * \
0 & * & 0 & * \
0 & * & 1 & *
endpmatrix,
endalign*
where $*$ can assume any real number. It is also clear for any monic $4^th$ degree real polynomial, we can at least find one realization in $star$ since the upper left block and lower right block can be considered as in companion form. We further concern this set
beginalign*
mathcal E = A in star: max_i left( lambda_i(A) right) < 0 .
endalign*
In other words, all elements in $mathcal E$ has above defined structure and all eigenvalues on the left open half plane. I am trying to determine whether the set $mathcal E$ is connected.

My first try was to determine for a fixed monic polynomial, whether all realizations in $star$ is connected. If this is true, for any $A_1, A_2 in mathcal E$, we may first connect them to a companion form in $star$ (by making the $32$ entry to be $1$ and other entries in the second column to be $0$) without changing the eigenvalues, and then the companion forms are connected as a consequence of property of polynomials. But as the question I asked a while ago, this is only true if it has at least one real eigenvalue.

I strongly believe the set is connected. The question is related but I am not sure it is that related. Because the condition there is much more strict. I was asking to connect all realizations in $star$ yielding the same characteristic equation without changing its eigenvalues. Here we allow this to vary inside $mathcal E$.

One may prove the set is connected by stability analysis of matrices. In following, I will concern the transpose of $star$ and this does not change the connectedness.

Note $star approx mathbb R^2 times 4$. Putting $A= pmatrix0 & 1 & 0 & 0\ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0$ and $B = pmatrix0 & 0 \ 1 & 0 \0 & 0 \ 0 & 1$, the set $mathcal E$ can be identified by a subset $mathcal F subset mathbb R^2 times 4$ with $mathcal F = X subset mathbb R^2 times 4: A-BX text is stable$.(see definition of stable matrix.) Now by Lyapunov matrix theory, $X in mathcal F$ if and only if there exists some positive definite matrix $P > 0$, such that $(A-BX) P + P(A-BX)^T < 0$ ($< 0$ means negative definite.) A change of variable $Y=XP$ yields
beginalign
labeleq:1
tag$dagger$
AP + PA^T - BY - Y^TB^T < 0.
endalign
Observe the solution set $(P, Y)$ of $dagger$ is a convex set and thus connected. So $mathcal F$ is a continuous image under the map $(P, Y) mapsto YP^-1$.

This method does not reveal too much of set structure which I think should be very nice. Hope someone can give a more transparent argument.