Non-square orthogonal matrix $B$ with $B^TB=I$

Question

Let $B\in\mathbb{R^{p\times q}},p\geq q$ such that $B^TB=I_q$, is it true that all the diagonal elements of $BB^T$ are all no greater than 1? How can I prove that?

Answer 1

$P=BB^T$ is an orthogonal projector, $P=P^T$, $P=P^2$. The diagonal elements satisfy $$ p_{nn}=e_n^TPe_n=e_n^TP^2e_n=\sum_{m=1}^kp_{mn}^2. $$ So you get $$ p_{nn}(1-p_{nn})=\sum_{m\ne n}^kp_{mn} $$ so the left side has to be non-negative.