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Let $V$ denote a nonzero finite-dimensional vector space over the complex field $\mathbb{C}$. Given a linear transformation $A:V\rightarrow V$, show that i) implies ii):

i) There exists an invertible linear transformation $P:V\rightarrow V$ such that $AP=-PA$

ii) There exists a direct sum decomposition $V=V_1\oplus V_2$ s.t. $AV_1\subset V_2$ and $AV_2\subset V_1$.

Eigenvalues of $A$ are $\pm$ pairs.

$V_1$ and $V_2$, in particular their bases, are probably to be guessed.

Please give a hint. Please do not give solution. Thanks!

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    $\begingroup$ Let $Av_+=\lambda v_+$ and $Av_-=-\lambda v_-$, and consider $v_++v_-$ and $v_+-v_-$. $\endgroup$ – Gerry Myerson Aug 7 '20 at 7:01
  • $\begingroup$ @GerryMyerson thanks. $\endgroup$ – Vinay Deshpande Aug 7 '20 at 12:12
  • $\begingroup$ If $A^kv=0$ for some positive integer $k$, consider the images of $v$ under the even powers of $A$ and the images of $v$ under the odd powers of $A$. $\endgroup$ – user1551 Aug 7 '20 at 13:16
  • $\begingroup$ @user1551 Ok nice. But that's not the case here. $\endgroup$ – Vinay Deshpande Aug 7 '20 at 13:36
  • $\begingroup$ Why not? Every nilpotent linear operator on $V$ is similar to its negative. $\endgroup$ – user1551 Aug 7 '20 at 13:36

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