Characterising orthogonal projectors

ProofIntermediate⏱ 30 minFoundationsprojectionidempotentsymmetric

Problem

Let $P \in \R^{d\times d}$ . Prove that the following are equivalent.

$P = P^\top$ and $P^2 = P$ .
There is a subspace $V \subseteq \R^d$ such that $Px$ is the closest point of $V$ to $x$ , for every $x$ .

Deduce that the eigenvalues of an orthogonal projector are all $0$ or $1$ , and that $\operatorname{tr}(P) = \dim V$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

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