# para-complex structure [closed]

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 $K=\pmatrix{ & Id_n \cr Id_n & \cr}$ ?
-

## closed as off topic by Dmitri Pavlov, Qfwfq, Misha, Andreas Blass, Qiaochu YuanMay 12 '13 at 18:55

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

If you want the question deleted, you should ask The User for permission. –  S. Carnahan May 31 at 10:24
$x_i=e_i+f_i, y_i=e_i-f_i$ where $e_1,\ldots,e_n,f_1,\ldots,f_n$ is the standard basis? Or what do you mean?
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