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In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then there exists an orthogonal matrix $P$ such that $PSP^{-1}$ is block diagonal with blocks of size at most two. While we have a proof of this statement, we are wondering whether it already appears in the literature. Have you seen this result or its generalizations published anywhere?

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    $\begingroup$ I think this is the kind of thing that is proven over-and-over inside lemmas and so on, insofar as I can see that it would follow from not-quite-intro-level linear algebra... thus, probably not graduating to the status even of "lemma" in any noticeable contemporary literature. So, I'd bet there's no reasonable cite-able source in contemporary mathematics. $\endgroup$ – paul garrett Feb 26 '16 at 0:26
  • $\begingroup$ I don't know, it could be an exercise in some textbook, although it could be just a touch too tricky for that. $\endgroup$ – Lev Borisov Feb 26 '16 at 1:11
  • $\begingroup$ Just as a comment: in the case of $S$ being an involution, the statement is pretty much equivalent to the principal angles between subspaces story. $\endgroup$ – Lev Borisov Feb 26 '16 at 17:37
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    $\begingroup$ Did you look at Theorem 6.4.14 of: ebooks.cambridge.org/ebook.jsf?bid=CBO9780511840371 $\endgroup$ – Suvrit Feb 27 '16 at 17:42
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    $\begingroup$ I found the relevant page online, it's not the right result. It is basically trivial to prove the statement without the orthogonality assumption on $P$. And it is not particularly hard to prove it with the orthogonality condition, but so far I didn't see a reference. $\endgroup$ – Lev Borisov Feb 27 '16 at 18:40

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