Equality of orthogonal projection norms.

Question

Let $F,G$ be two sub vector spaces of $\mathbb{R}^n$, and denote $P_F,P_G$ the orthogonal projection matrices on $F$ and $G$ respectively. Suppose that for all $v \in \mathbb{R}^n$ we have: \begin{equation*} ||P_Fv||^2 = ||P_Gv||^2 \end{equation*} where $||\cdot||$ is the euclidean norm. Is it true that one of the following assertion holds?

i) $F \subset G$

ii) $G \subset F$

iii) F=G

Answer 1

From $||P_Fv||^2 = ||P_Gv||^2$ for all $v \in \mathbb R^n$ we get

$v \in F^{\perp} \iff P_Fv=0 \iff P_Gv=0 \iff v \in G^{\perp} .$

Hence $F^{\perp}=G^{\perp}$.

Can you proceed ?