Complex structure with a linear transformation

Question

A complex structure on a real vector space $V$ is a linear endomorphism $J$ of $V$ such that $J^2=−1$, where $1$ is the identity transformation of $V.$ Let $T: V \rightarrow V$ be a linear transformation such that $T$ commutes with $J.$ Prove that det $T>0$

Answer 1

I assume you mean $T$ isn't the zero transformation.

Hint: Here's the answer for $V = \mathbb R^2$, outlined in the steps below.

1. $J$ necessarily(see wikipedia or this question) has the form $\begin{bmatrix} a & b\\ c & -a \end{bmatrix}$, with determinant = 1.

2. $T$ commutes with $J$ if and only if $T = xI_2+yJ$ for some $x,y \in \mathbb R$.

   • First note that $c \ne 0$. Then, for $T = \begin{bmatrix} j & f\\ g & h \end{bmatrix}$, choose $x=j-ay$ and $y=\frac gc$.

3. By Steps 1 and 2, $\det T = x^2 + y^2$. This is always nonnegative, and is positive if and only if $T$ is not the zero transformation.