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Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on the dimension?

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Let $B$ be an invertible skew-symmetric matrix of order $2n$.

Let $C=\left(\begin{array}{cc} B & 0_{2n\times 2n} \\ 0_{2n\times 2n} & B \end{array}\right)$. Notice that $C$ is also skew-symmetric and invertible.

Let $R$ be any orthogonal matrix such that $RBR^{-1}\neq B$.

Let $D=\left(\begin{array}{cc} R & 0_{2n\times 2n} \\ 0_{2n\times 2n} & Id_{2n\times 2n} \end{array}\right)$. Notice that $C$ is also orthogonal.

Now $DCD^{-1}-C=\left(\begin{array}{cc} RBR^{-1}-B & 0_{2n\times 2n} \\ 0_{2n\times 2n} & B-B \end{array}\right)=\left(\begin{array}{cc} RBR^{-1}-B & 0_{2n\times 2n} \\ 0_{2n\times 2n} & 0_{2n\times 2n} \end{array}\right)$.

Notice that $DCD^{-1}-C\neq 0$ and is not invertible.

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