Let $A$, $B$ be $n\times n$ unitary complex matrices, such that for all indices $i,j$ we have $|a\_{ij}|=|b\_{ij}|$. Does there then exist diagonal unitary matrices $D,D’$ such that $DAD’=B$?


This can also be phrased as, take a unitary matrix, and sprinkle norm one signs on all of its entries. If the resulting matrix is unitary, then your sign at $(i,j)$ is equal to $z\_i\cdot w\_j$ for some lists of norm one complex numbers $(z\_i)\_{i=1}^n$, $(w\_j)\_{j=1}^n$.


The unitary case implies the analogous result for orthogonal matrices, and the positivity here makes the problem feel much easier. I would also be interested in a proof for this orthogonal case.


I have checked this for some small $n$, and the orthogonal case has the feeling of a combinatorial problem on intersecting families of sets, but I wasn’t able to make precise this intuition.



Sprinkling signs in unitary matrices
There are [5 inequivalent Hadamard matrices](https://en.wikipedia.org/wiki/Hadamard_matrix#Equivalence_and_uniqueness) of order 16; if I understand correctly that's a counterexample.


