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Definition :Let $V$ be a finite dimensional real vector space. A para-complex structure on $V$ is an endomorphism $K$ : $V \to V$ such that:

  1. $K$ is an involution, that is $K^2 = Id_V$ ;
  2. The eigenspaces $V := ker(Id_V \mp K)$ of $K$ with eigenvalues $1$ respectively have the same dimension. A vector space $V$ endowed with a para-complex structure $K$, denoted by $(V;K)$, will be called para-complex vector space. We know, we can identify $K$ with $K=\pmatrix{Id_n & \cr & -Id_n\cr} $ . Is there a basis of vectors like {$ x_1,x_2,...,x_n,y_1,y_2,...,y_n $}, such that $Kx_i=y_i$ and $Ky_i=x_i$. i.e. can we write is such basis $K=\pmatrix{ & Id_n \cr Id_n & \cr} $ ?
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closed as off topic by Dmitri Pavlov, Qfwfq, Misha, Andreas Blass, Qiaochu Yuan May 12 '13 at 18:55

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1 Answer 1

up vote 3 down vote accepted

$x_i=e_i+f_i, y_i=e_i-f_i$ where $e_1,\ldots,e_n,f_1,\ldots,f_n$ is the standard basis? Or what do you mean?

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I guess that this is too trivial—could you clarify? –  The User May 10 '13 at 22:59
Btw, you are using $V$ for multiple different spaces I guess. –  The User May 10 '13 at 23:02
Yes, we just need this trick. because $Ke_i=e_i$ and $Kf_i=-f_i$ so we get the desired result.thanks "The User". –  Hassan Jolany May 10 '13 at 23:04

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