  We say that a list of complex numbers is "realisable" if it is the spectrum
of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue
Problem (NIEP) is the problem of characterising all realisable lists. Although
the NIEP remains unsolved, it has been solved in the case where every entry in
the list (apart from the Perron eigenvalue) has nonpositive real part. For a
given spectrum of this type, we show that a list of nonnegative numbers may
arise as the diagonal elements of the realising matrix if and only if these
numbers satisfy a remarkably simple inequality. Furthermore, we show that
realisation can be achieved by the sum of a companion matrix and a diagonal
matrix.
