The NIEP

TL;DR

This survey reviews the extensive research on the nonnegative inverse eigenvalue problem (NIEP), highlighting recent advances, various subproblems, and key theoretical developments in understanding which eigenvalues can be realized by nonnegative matrices.

Contribution

It provides a comprehensive overview of the NIEP, emphasizing recent results and organizing the literature into multiple thematic categories with 130 references.

Findings

Summarizes necessary and sufficient conditions for eigenvalue realizability.

Highlights progress in low-dimensional cases and graph-based variants.

Discusses the role of Jordan structure and Perron similarities in the problem.

Abstract

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.