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Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that $A=QBQ^{−1}$?)?

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No.

Consider $A= \left[\begin{array}{cc} 0 & 1 \\0 & 0 \end{array}\right]$, $B=\left[\begin{array}{cc} 0&0\\-1&0\end{array}\right]$ and $P = \left[\begin{array}{cc} 0&1\\-1&0\end{array}\right]$. Then $A = PBP^{-1}$.

But for a symmetric $Q=\left[\begin{array}{cc} a&b\\b&c\end{array}\right]\in GL_n(\mathbb R)$. If $$ A = QBQ^{-1} = \frac{1}{ac - b^2}\left[\begin{array}{cc} -bc & b^2 \\ -c^2 & bc\end{array}\right]. $$ Then $c=0$ which implies $$ A = \left[\begin{array}{cc} 0&-1\\0&0 \end{array}\right], $$ a contradiction.

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