Given an orthogonal matrix $U$, i.e., $U^TU = I$ and diagonal matrices $P, Q$ such that
$ A = P U Q $
All matrices are real, and $I$ denotes the identity matrix.
For which $P$ and $Q$ is there a vector $b$ such that $ I - A A^T = bb^T $ ?
For example, $Q = I$ and $P$ almost identity except for a single diagonal entry, yield a solution for $b$. The roles of $P$ and $Q$ can be switched. What are the other solutions?
Your condition is equivalent to $spectrum(PUQ^2U^TP)=\{1,\cdots,1,u\}$ where $u\in [0,1]$.
EDIT. We can also write the condition in algebraic form.
Let $p$ be the characteristic polynomial of $PUQ^2U^TP\in M_n(\mathbb{R})$. Then the NS condition on $p$ is: $p$ has $1$ as root of multiplicity $n-1$ and the product of its roots is $\leq1$, that is:
$p(1)=p'(1)=\cdots=p^{(n-2)}(1)=0$ and $|\det(P)\det(Q)|\leq 1$.