In my course of Linear Algebra we were assigned the following problem:
Let $V$ be a vector space over the field $\mathbb F = \mathbb R$ or $\mathbb F=\mathbb C$ with $\dim V<\infty$ and let $X \in \mathrm{O}(V)$ (i.e. an orthogonal matrix). Also, let $J$ be a linear operator on $V$ such that $J^2=-1$.
(a) Prove that the linear transform $\iota:X \mapsto -JXJ$ is an involution; that is, that $\iota^2 X=X$ for all $X \in \mathrm{O}(V)$.
(b) Define $T(X)=\frac{X-JXJ}{2}$. Prove that $T$ is a projection into $E_1$.
Note: $E_{-1}$ and $E_1$ are the eigenspaces associated to the eigenvalues $\lambda=—1$ and $\lambda=1$, respectively.
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Part (a) of this problem is very straightforward, I just computed $\iota^2(X)$ and used the definition of $J$.
My problem is in part (b): I notice that $T(X)=\dfrac{1}{2}(X+\iota(X))$, but for some reason I can’t achieve the equality $T^2=T$.
And even if I manage to do that then how do I know that $T$ is a projection into $E_1$? Do I need to find explicitly the set $E_1$?
How can I find explicitly (if possible) the vector subspaces $E_{\pm 1}$? Since the orthogonal matrix $X$ is arbitrary then $\iota$ is also arbitrary.
Thanks a lot in advance.