The geometry of the set of scaled projections (Q686373)

Let \(A\) be a full column rank matrix and \(b\) a fixed vector not in the range of \(A\). Every projection onto the range of \(A\) has a representation \(P_ W=A(A^ TWA)^{-1}A^ TW\) where \(W\) is symmetric and positive definite. The set of scaled projections of \(b\) is the set of all \(P_ Wb\) where \(W\) is a positive diagonal matrix. In this paper it is shown that the set of scaled projections of \(b\) is the union of finitely many polytopes.