I want to confirm if a projector matrix is its own inverse. I have $x=Px$ and $Px=P^2x$, so premultiplying the second equation with $P^{-1}$ twice, I get $P^{-1}x=Px$ for all x, implying $P^{-1}=P$. Is this reasoning correct?
So are all projection matrices orthogonal too?
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$\begingroup$No, this reasoning isn't correct because you assumed that the projection matrix has an inverse without proving it. In fact a projection matrix is a good example of a matrix that doesn't have an inverse: Part of the vector it is applied to is projected out, and there's no way to reconstruct that part.
$\endgroup$ $\begingroup$Projector matrices are idempotent, and as a rule, need not have an inverse at all (since it will usually have a non-trivial nullspace). For $P$ to be its own inverse, we need $P^2=I$. Since $P^2=P$ for any projector matrix, then the only projector matrix that is its own inverse is the identity (which we can think of as the trivial projector of a space onto itself).
$\endgroup$ 2 $\begingroup$If by "projector matrix" you mean the matrix of a projection onto a (proper) subspace, then the rank of such a matrix will be the dimension of that subspace, which is less than the number of columns. So the nullspace is nontrivial, and the matrix is not invertible.
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