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One of Jordan's lemma states that if two orthogonal matrices $A,B$ are such that $A^2=B^2=I$, then they can be co-diagonalized by block of size 2. (the proof is easy, consider $x$ an eigenvector of $A+...

Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...