For context, this question comes from my reading of Kobayashi+Nomizu's differential geometry book, volume 2 (pages 116-117).
Given a real vector space $V$ with $\mathrm{dim}(V)=2n$, a complex structure is a linear endomorphism satisfying $J^2=-1$.
One can also define the complexification of a vector space as $V^{\mathbb{C}} = V \otimes_{\mathbb{R}} \mathbb{C}$.
Now, $J$ extends naturally to a complex endomorphism on the complexified space, and (Kobayashi and Nomizu claim that) it has eigenvalues $\pm i$.
I have some confusions about this:
- Is it true that the natural extension of $J$ is just defined by $J(V\otimes z) = J(V)\otimes z$?
- More crucially, the complexified vector space is treated as a tensor product over the reals, so isn't the coefficient field of this new space also the reals? Isn't, then, the characteristic polynomial viewed as a polynomial over $\mathbb{R}$ (and thus cannot have imaginary roots)?
- My intuition is to treat application of $J$ as multiplication by $i$ in the original space, but the complexified space offers a bona fide multiplication by $i$, and I'm not sure how these two interact. What would be an example of a $v\in V^{\mathbb{C}}$ such that $Jv = -iv$? What does $Jv=iv$ actually mean? I've tried to answer these questions for $V=\mathbb{R}^2$ with the canonical structure but I'm coming up empty handed.
I greatly appreciate your time and help!