Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in GL_n(\mathbb{C})$ such that $$ G_i = T^{-1} F_i T $$ for all $i \in [k]$. Does there exist $V \in U_n$, where $U_n$ is the group of unitary matrices, such that $$ G_i = V^{-1} F_i V $$ for all $i \in [k]$?

**Added later:**

Generally, let $\pi, \sigma : G \to U_n(\mathbb{C})$ be two unitary representations of a finite group. Suppose $\pi$, $\sigma$ are equivalent, i.e. there exists $T \in GL_n(\mathbb{C})$ such that $T \pi(g) = \sigma(g) T$ for all $g \in G$. Are $\pi$ and $\sigma$ unitarily equivalent, that is, there exists $T' \in U_n$ such that $T' \pi = \sigma T'$?

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