A unified view on compensation criteria in the real nonnegative inverse eigenvalue problem

A connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices (the so-called real nonnegative inverse eigenvalue problem) and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers. Given realizable sets (i.e., sets which are spectra of some nonnegative matrix), each of these three elementary transformations constructs a new realizable set. This defines a special kind of realizability, called C-realizability and this is closely related to the idea of compensation. After observing that the set of all C-realizable sets is a strict subset of the set of realizable ones, we show that it strictly includes, in particular, all sets satisfying several previously known sufficient realizability conditions in the literature. Furthermore, the proofs of these conditions become much simpler when approached from this new point of view.