I’m having a little bit of trouble with a homework assignment of my Linear Algebra course. Here it is the statement:
Let $V$ be a real (or complex) vector space of finite dimension and let $X \in \mathrm{O}(V)$. Define the invertible linear transformation $J:V \to V$ such that $J^2=-I$ and $$J^{-1}=J^t=-J.$$ Also, define the conjugation map $\phi$ as $$\phi:X \to JXJ^{-1}=-JXJ.$$
(a) Prove that $\phi$ is an involution map. Moreover, prove that any $X \in \mathrm{O}(V)$ is can be expressed as as sum of the type $$X=K_+(X) + K_{-}(X),$$ where you need to find, explicitly, the matrices $K_{\pm}(X)$.
(b) Prove that $V= \ker K_+(V) \oplus \ker K_{-}(V)$.
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Part (a) is pretty straightforward: Note that $$\phi^2(X)=\phi(\phi(X))=-J\phi(X)J=-J(-JXJ)J=(-1)^2J^2XJ^2=(-1)^2X(-1)^2=X$$ since $J^2=-I$. Also, note that $$X=\frac{X+\phi(X)}{2}+\frac{X-\phi(X)},$$ so $K_{\pm}(X)= \frac{X \pm \phi(X)}{2}$.
But I’m having trouble with part (b) of the problem: Since $X$ is an orthogonal matrix then $\det(X)=\pm 1$ and therefore $X$ is invertible. And since $X$ is invertible, both $\ker(K_+(X))$ and $\ker(K_{-}(X))$ are (linearly) independent subsets of $V$.
The problem I’m having is that in order for the sum $V=\ker(K_{+}(X)) + \ker(K_{-}(X))$ to be direct, the kernels must orthogonal complements, but I haven’t bern able to prove this. I assume (since I haven’t use that hypothesis) that $J$ is orthogonal, but I don’t know how to proceed. Or am I doing something wrong?
Any help will be greatly appreciated, thank you!