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!