What is the determinant of a weighted orthogonal projection (based on the weighted pseudo-inverse)? What is the determinant of a weighted orthogonal projection (based on the weighted pseudo-inverse)? E.g. I have 
$$ J = A \left( A^\intercal W A \right)^{-1} A^\intercal W $$
and would like to know $\det (J)$. Note that $A$ is not square while $W$, $J$, and $A^\intercal W A$ are square.  

The kernel of any projection onto a proper subspace is nontrivial. If $A$ is not square, then its columns don’t span the entire ambient space, therefore $J$ is rank-deficient and $\det(J)=0$.
