Complex structures vs inner products on $\mathbb{R}^2$

Question

A complex structure on $V := \mathbb{R}^2$ is a linear transformation $J : V\rightarrow V$ satisfying $J^2 = -1$.

If $B(\cdot,\cdot)$ is an inner product on $V$, let $SO(V,B)$ be the subgroup of $GL(V)$ given by automorphisms $T$ satisfying $B(Tv,Tv) = B(v,v)$ for all $v\in V$.

If $V := \mathbb{R}^2$ is given the standard inner product given by the dot product "$\cdot$", then $SO(V,\cdot)$ is just the circle group, and there seems to be exactly two complex structures $J$ which preserve the dot product, corresponding to rotating by $\pm\pi/2$.

For an arbitrary inner product $V$ on $V$, how can we write down the complex structures on $V$ preserving $B$?

If we think of $B$ as a symmetric positive definite $2\times 2$ matrix, then this seems to amount to solving the equations $J^2 = -1, J^tBJ = B$. It's unclear what we can read off from this equation. For example, is it clear (or even true) that there are always exactly two solutions $J$?

Conversely, suppose we are given a complex structure $J$ on $V$. Can we classify the inner products $B$ which are preserved by $J$?

Again, we want to solve $J^tBJ = B$, this time for $B$. Certainly if $B$ is preserved by $J$, then so is $cB$ for any $c\in\mathbb{R}$. Is it clear that the solution set of $B$ satisfying $J^tBJ = B$ is always 1-dimensional?

Answer 1

Looking at this geometrically will inform the generalization, I think. The condition $B(v,v) = B(Tv,Tv)$ says that the norm induced by $B$ is preserved, i.e., that the related quadratic form $\mathbf q(v)=B(v,v)$ is invariant under $T$. The level curves of $\mathbf q$ are a family of homothetic ellipses, therefore you’re looking for an automorphism of the ellipse $\mathbf q(v)=1$. Will Jagy explains here that these automorphisms are conjugate rotations and reflections $M^{-1}RM$, where $M$ is chosen so that (with some abuse of notation) $M^{-T}BM^{-1}=I$. (I.e., $M$ maps between the unit circle and the ellipse $\mathbf q(v)=1$.) It seems reasonable to expect, then, that the complex structures admitted by $B$ are the corresponding conjugate rotations by $\pm\pi/2$. This makes geometric sense: a conjugate rotation by $\pm\pi/2$ maps a point on the ellipse to an endpoint of the conjugate diameter, and applything this rotation twice takes the point to its diametric opposite as required.

In a very real sense, the choice of an inner product $B$ induces a notion of angle in the vector space via the usual identity $B(v,w)=\mathbf q(v)^{1/2}\mathbf q(w)^{1/2}\cos\theta$. From this point of view, the conjugate rotations above are the rotation-by-$\theta$ operators on the space. In projective-geometric terms, choosing an inner product is equivalent to fixing the circular points $\mathtt I$ and $\mathtt J$, which induces a Euclidean geometry on the projective plane.

Starting instead with the other constraint $J^2=-I$, it’s not hard to see that the matrix of $J$ must be similar to the standard rotation-by-$\pi/2$ matrix. The possible eigenvalues of $J$ are $\pm i$, but none of the cases with a repeated eigenvalue lead to real-valued matrices. So, the potential complex structures are precisely the conjugate ninety-degree rotations. If $J$ has the matrix $M^{-1}RM$, then a corresponding inner product has the matrix $M^TM$.