Suppose $A$ and $B$ are distinct permutation matrices. I'm trying to identify some properties of $A-B$ regarding the positions and values of nonzero entries in this matrix. Maybe even characterizations. Orthogonality, for example, is discarded because $A-B$ does not necessarily have this property.
One "invariant" I find in examples is that every nonzero column/row of $A-B$ has exactly one $-1$ and one $1$. For my study, this is very pertinent, but I don't know to formalize it properly.
In addition, if I include the hypothesis that $A-B$ is symmetric, there are some restrictions for the positions of the nonzero entries, but which exactly?
In summary, could you point me to some literature regarding this topic? What comes to your mind if you think about the difference between permutation matrices?
Apologies if the questions are somewhat vague or basic. I'm a beginner in linear algebra.