Is difference of two projection matrices positive semi-definite or negative definite or indefinite?

Question

Suppose $S$ is an $m \times n$ matrix of full column rank and $W$ is an $m \times m$ positive definite matrix.

Let $R = W^{-1/2}S$ and $Q = W^{1/2}S$.

What can we say about $R(R^\top R)^{-1}R^\top - Q(Q^\top Q)^{-1}Q^\top$? Is it positive semi-definite or negative definite or indefinite?

Answer 1

If the result is positive semi-definite for some $$W=W_1\succ0,$$ then it will be negative semi-definite for $$W=W_1^{-1} \succ 0,$$ since $R$ and $Q$ will simply switch places.

Answer 2

Presumably the matrices are real. The difference in question is $RR^+-QQ^+$, which is a difference of two orthogonal projections. Therefore \begin{align} &RR^+-QQ^+\preceq0\\ \Leftrightarrow\ &\operatorname{range}(RR^+)\subseteq\operatorname{range}(QQ^+)\\ \Leftrightarrow\ &\operatorname{range}(R)\subseteq\operatorname{range}(Q)\\ \Leftrightarrow\ &W^{-1/2}\operatorname{range}(S)\subseteq W^{1/2}\operatorname{range}(S)\\ \Leftrightarrow\ &W^{-1/2}\operatorname{range}(S)=W^{1/2}\operatorname{range}(S)\ \text{(because both sides have equal dimensions)}\tag{1}\\ \Leftrightarrow\ &\operatorname{range}(S)=W\operatorname{range}(S).\tag{2} \end{align} However, by a similar argument, we also have $RR^+-QQ^+\succeq0$ if and only if $(1)$ holds. Hence $RR^+-QQ^+$ is either both positive semidefinite and negative semidefinite, or indefinite. In other words (and by $(2)$), $RR^+-QQ^+$ is zero when $\operatorname{range}(S)$ is an invariant subspace of $W$, or indefinite otherwise.