I am extracting information of me proving an application problem, if there is anything not clear, please let me know.
Suppose we parameterize a structured matrix in following form \begin{align} \label{matrix:1} \tag{$\star$} \begin{pmatrix} 0 & -a_1 & 0 & 0 & -b_1 \\ 1 & -a_2 & 0 & 0 & -b_2 \\ 0 & -a_3 & 0 & 0 & -b_3 \\ 0 & -a_4 & 1 & 0 & -b_4 \\ 0 & -a_5 & 0 &1 & -b_5 \end{pmatrix}, \end{align} where the upper left $2 \times 2$ and lower right $3 \times 3$ blocks are in companion form.
I would like to claim that the set of matrices of above form and with eigenvalues modules bounded in some interval $(-M, M)$ is connected. If we let \begin{align*} E = \{ A \in \mathcal{M}(5 \times 5; \mathbb R): \text{ $A$ is of form \eqref{matrix:1} and } \rho(A) < M \}, \end{align*} we want to prove $E$ is connected.
My approach is to first prove the set \begin{align*} F = \{ B \in \mathcal{M}(5 \times 5; \mathbb R): \text{ structure of $B$ is of the form \eqref{matrix:2} and } \rho(B) < M \} \end{align*} is connected, where \begin{align} \label{matrix:2} \tag{$\ast$} \begin{pmatrix} 0 & -a_1 & 0 & 0 & 0 \\ 1 & -a_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -b_3 \\ 0 & 0 & 1 & 0 & -b_4 \\ 0 & 0 & 0 &1 & -b_5 \end{pmatrix}. \end{align} If a matrix $C$ has the form \eqref{matrix:2}, then the characteristic polynomial $p_C(t) = (t^2+a_2 t + a_1)(t^3 + b_5 t^2 + b_4 t +b_3)$. If I am not mistaken, the set $F$ should be a continuous image of roots lying in $(-M, M)$ which should be connected. Now I would like to show for a general element in $E$, we can find a continuous path to the set $F$. Since for any $A \in E$, the characteristic polynomial $$p_A(t) = (t^2+a_2 t + a_1)(t^3 + b_5 t^2 + b_4 t +b_3) + \text{ polynomial terms involving entries of } a, b.$$ Intuitively, we can first choose some element $D \in F$ with $p_D(t) = p_A(t)$ and then continuously transform the coefficients there. But now I could not prove the whole path will only contain elements of $E$.
Would anyone help me with this approach or provide some other approaches? Or given examples to show the set is actually not connected?
Let $r\ne 0$ and consider the matrix $$ A_r:=\begin{pmatrix} 0 & - r^2 a_1 & 0 & 0 & - r^2 b_1 \\ 1 & - r a_2 & 0 & 0 & - r b_2 \\ 0 & - r^3 a_3 & 0 & 0 & - r^3 b_3 \\ 0 & - r^2 a_4 & 1 & 0 & - r^2 b_4 \\ 0 & - r a_5 & 0 &1 & - r b_5 \end{pmatrix}. $$ Then the characteristic polynomial $P_{A_r}$ of $A_r$ satisfies
$$ P_{A_r}(rx)=r^5 P_A(x). $$ Hence, $\lambda$ is an eigenvalue of $A$ if and only if $r\lambda$ is an eigenvalue of $A_r$.
Now it is easy to connect two matrices $A_0,A_1$ in $E=E_M$:
Take the continuous path $A_t=tA_1+(1-t)A_0$ from $A_0$ to $A_1$. Since the path is compact, there is an $N>0$ such that $\rho(A_t)<N$ for all $t$. If $N\le M$, the path lies in $E$. Else set $r:=M/N$ and connect $A_0$ with $(A_0)_r$ via $(A_0)_s$ for $s\in[r,1]$, connect $(A_0)_r$ with $(A_1)_r$ via $(A_t)_r$ for $t\in [0,1]$ and connect $(A_1)_r$ with $A_1$ via $(A_1)_s$ for $s\in[r,1]$. The three paths, concatenated, give the desired path between $A_0$ and $A_1$ in $E$.