The nonnegative inverse eigenvalue problem (NIEP) asks which
lists of n complex numbers (counting multiplicity) occur as the eigenvalues of
some n-by-n entry-wise nonnegative matrix. The NIEP has a long history and
is a known hard (perhaps the hardest in matrix analysis?) and sought after
problem. Thus, there are many subproblems and relevant results in a variety
of directions. We survey most work on the problem and its several variants,
with an emphasis on recent results, and include 130 references. The survey
is divided into: a) the single eigenvalue problems; b) necessary conditions; c)
low-dimensional results; d) sufficient conditions; e) appending 0’s to achieve
realizability; f) the graph NIEP’s; g) Perron similarities; and h) the relevance
of Jordan structure.
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