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Given any triple of $n \times n$ matrices $(A, B, J)$ and invertible $2n \times 2n$ matrix $U$ for which $U(J \oplus -J)U^{-1} = \begin{pmatrix}0 & A \\ B & 0\end{pmatrix}$, can we find a pair of $n \times n$ invertible matrices $(P,Q)$ such that $\begin{pmatrix}-P & P \\ Q & Q \end{pmatrix} (J \oplus -J)\begin{pmatrix}-P^{-1} & Q^{-1} \\ P^{-1} & Q^{-1}\end{pmatrix} = 2\begin{pmatrix}0 & A \\ B & 0\end{pmatrix}$?

Note: All the matrices are over $\mathbb C$. $\oplus$ denotes direct sum. The names of the matrices don't imply anything about the matrices that isn't stated in the question.

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  • $\begingroup$ What might the names of matrices imply that they don't actually? $\endgroup$
    – LSpice
    Commented Nov 2, 2021 at 19:43
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    $\begingroup$ @LSpice $U$ is usually used for unitary matrices and $J$ for Jordan matrices. But neither condition has to hold. $\endgroup$
    – wlad
    Commented Nov 2, 2021 at 19:46

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