connectedness of conjugacy classes of a fixed matrix $A$ but with the first column invariant

Question

Let $A \in M_n(\mathbb C)$ be a fixed matrix. The set $\{S^{-1} A S: S \in GL_n(\mathbb C)\}$ is a continuous image of $GL_n(\mathbb C)$, so I think it should be a connected set. Let $A = (a_1, \dots, a_n)$ where $a_j \in \mathbb C^n$ denotes the columns of $A$. Let $F = \{S^{-1} A S: S \in GL_n(\mathbb C) \text{ and } (S^{-1}AS)_{\cdot,1} = a_1\}$. Is the set $F$ still connected?

Answer 1

Your condition $(S^{-1}AS)_{\cdot,1} = a_1$ is equivalent to $(SA-AS)e_1 = 0$. Let $S$ be a matrix in your set and let $\gamma : [0,1]\to\mathbb C$ be a path such that $\gamma(0) = 0$, $\gamma(1) = 1$, and $\det(S + \gamma(t)(I-S))\neq 0$. It is now easy to see that $S(t) = S + \gamma(t)(I-S)$ is in your set for each $t\in [0,1]$ with $S(0) = S$ and $S(1) = I$. So, we have found a path within your set from $S$ to $I$. This shows that it is connected.