Why is oblique projection not a self adjoint operator? Why is oblique projection not a self adjoint operator? Here is an explanation of oblique projection.

I think the reason is:

we want $\ \forall x,y \ = $

If $P$ is orthogonal projection, then this is due to $$\ =\ \ = \ $$ and the latter is due to the decomposition $ y = Py + y^*$, so that for every $z$, $ = 0$. (That is, $y^*$ is orthogonal to the hyperplane we're projecting to.) Therefore, $\ =\ $.

But if your projection is not orthogonal, then $ \
e 0$ and the above calculation fails.

(Try to picture it in 2 dimensions.)