Positive semi-definiteness of the difference between oblique and orthogonal projections

Question

Suppose $P = X(X^\top W^{-1}X)^{-1}X^\top W^{-1}$ and $Q = X(X^\top X)^{-1}X^\top$.

Under what conditions: $QVQ^\top - PVP^\top$ is positive semidefinite? $V$ is positive semi-definite.

Is it only when $V = W$?

Answer 1

No, take $W = I$ and $V$ any positive semi-definite matrix. In that case, $P V P^T - Q V Q^T$ is equal to the zero matrix.

Answer 2

Some discussions (To be continued)

Problem: Suppose $W$ is positive definite and $X$ has full column rank. Let $Q = X(X^TX)^{-1}X^T$ and $P = X(X^TW^{-1}X)^{-1}X^TW^{-1}$. Let $V$ be positive semi-definite. Under what conditions, $QVQ^T - PVP^T$ is positive semi-definite?

Two trivial cases: i) $X$ is a square matrix; ii) $W = c_1 I$ for some $c_1 > 0$.

In the following, assume that $X$ is not a square matrix, and $W \ne c_1 I$ for any $c_1>0$.

A simple case: $V = XAX^T + c_2W$ where $A$ is positive semi-definite and $c_2 \ge 0$.