Solution
= Solution
Because $\operatorname{col}Z\subseteq\operatorname{col}X$, the two <orthogonal projection matrices> satisfy
$$
P_0P=PP_0=P_0.
$$
Consequently $P-P_0$ is itself an orthogonal projection, onto the orthogonal complement of $\operatorname{col}Z$ within $\operatorname{col}X$. In particular it is a <positive semidefinite matrix>, so for every $u\in\mathbb R^n$,
$$
\boxed{u^TPu-u^TP_0u
=u^T(P-P_0)u
=\|(P-P_0)u\|^2\ge0}.
$$