Why is oblique projection not a self adjoint operator? Here is an explanation of oblique projection.
2026-03-26 12:33:43.1774528423
Why is oblique projection not a self adjoint operator?
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I think the reason is:
we want $\ \forall x,y \ <Px,y> = <x,Py> $
If $P$ is orthogonal projection, then this is due to $$<Px,y>\ =\ <Px,Py>\ = \ <x,Py> $$ and the latter is due to the decomposition $ y = Py + y^*$, so that for every $z$, $<y^*, Pz> = 0$. (That is, $y^*$ is orthogonal to the hyperplane we're projecting to.) Therefore, $<Px,y>\ =\ <Px,Py>$.
But if your projection is not orthogonal, then $<y^*, Px> \ne 0$ and the above calculation fails.
(Try to picture it in 2 dimensions.)