Matrix similar to its negative

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.

• Let $Av_+=\lambda v_+$ and $Av_-=-\lambda v_-$, and consider $v_++v_-$ and $v_+-v_-$. – Gerry Myerson Aug 7 '20 at 7:01
• 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 $A$. – user1551 Aug 7 '20 at 13:16
• Why not? Every nilpotent linear operator on $V$ is similar to its negative. – user1551 Aug 7 '20 at 13:36