Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.
Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.