Question about idempotent matrices.

Question

Let $E$ be the $m \times m$ matrix that extracts the "even part" of an $m$-vector $Ex = (x+Fx)/2$, where $F$ is the $m\times m$ matrix that flips $[x_1,\dotsc ,x_m]^{T}$ to $[x_m,\dotsc ,x_1]^T$. Is $E$ idempotent?

Answer 1

Yes, of course. What happens if you extract the "even part" of a vector which is already "even"?

$$E x=\left[\begin{matrix}(x_1+x_m)/2 \\ (x_2+x_{m-1})/2 \\ \vdots \\ (x_m+x_1)/2 \end{matrix}\right]$$ $$E^2 x=\left[\begin{matrix}((x_1+x_m)/2 + (x_m+x_1)/2)/2 \\ ((x_2+x_{m-1})/2 + (x_{m-1}+x_2)/2)/2 \\ \vdots \\ ((x_m+x_1)/2+(x_1+x_m)/2)/2 \end{matrix}\right]=\left[\begin{matrix}(x_1+x_m)/2 \\ (x_2+x_{m-1})/2 \\ \vdots \\ (x_m+x_1)/2 \end{matrix}\right]=E x$$