[Paper Review] The NIEP

Charles R. Johnson, Carlos Marijuán+2 more|arXiv (Cornell University)|Mar 31, 2017

Matrix Theory and Algorithms88 references4 citations

TL;DR

This comprehensive survey synthesizes over a century of research on the nonnegative inverse eigenvalue problem (NIEP), systematically organizing known results across eight thematic areas—ranging from single eigenvalue constraints to Jordan structure implications. It provides the most extensive reference compilation (130 sources) and offers updated insights into realizability conditions, sufficient criteria, and structural properties of nonnegative matrices, serving as a definitive resource for ongoing and future research in matrix analysis.

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.

Motivation & Objective

• To consolidate and systematize the vast body of research on the nonnegative inverse eigenvalue problem (NIEP), which remains one of the most challenging open problems in matrix analysis.
• To clarify the current state of knowledge by organizing results into eight thematic categories: single eigenvalue problems, necessary conditions, low-dimensional cases, sufficient conditions, zero-appending strategies, graph-based NIEP variants, Perron similarities, and Jordan structure relevance.
• To highlight recent advances and unresolved challenges in NIEP, particularly in identifying realizability criteria and structural constraints for nonnegative matrices.
• To serve as a comprehensive reference for researchers by compiling 130 key references and mapping the evolution of ideas across decades of research.

Proposed method

• The paper employs a thematic, categorical synthesis of existing literature, dividing the NIEP into eight distinct research domains to enhance clarity and navigability.
• It analyzes and compares known necessary conditions (e.g., trace and spectral radius constraints) and sufficient conditions (e.g., Brauer and Rado-type theorems) for eigenvalue lists to be realizable by nonnegative matrices.
• The authors examine low-dimensional cases (n ≤ 4) in detail, identifying exact realizability criteria and structural patterns not evident in higher dimensions.
• The paper investigates the effect of appending zeros to eigenvalue lists, establishing conditions under which realizability is preserved or achieved.
• It explores graph-theoretic variants of the NIEP, linking eigenvalue realizability to the structure of directed graphs and nonnegative matrix patterns.
• The role of Jordan structure in determining realizability is evaluated, particularly in relation to nonderogatory matrices and spectral properties.

Experimental results

Research questions

• RQ1 What are the complete sets of necessary and sufficient conditions for a list of complex numbers to be the spectrum of some nonnegative matrix?
• RQ2 How do low-dimensional cases (n ≤ 4) inform the general structure of solutions to the NIEP?
• RQ3 Under what conditions can a nonnegative matrix be constructed by appending zeros to a given eigenvalue list?
• RQ4 How do graph-theoretic constraints influence the realizability of eigenvalue lists in nonnegative matrices?
• RQ5 To what extent does the Jordan canonical form of a matrix constrain or enable its realizability as a nonnegative matrix?

Key findings

• The paper establishes that for n ≤ 4, complete and explicit realizability criteria are known, providing a foundational understanding of the NIEP in low dimensions.
• It confirms that appending zeros to a realizable list preserves realizability under certain spectral conditions, extending the scope of known realizability results.
• The survey identifies that Perron similarities—transformations preserving nonnegativity and spectral structure—are instrumental in constructing new realizable spectra from known ones.
• It demonstrates that Jordan structure significantly influences realizability, particularly in cases where nonderogatory matrices are required.
• The paper highlights that while many sufficient conditions exist (e.g., Rado’s theorem), no general necessary and sufficient condition is known for n > 4, underscoring the problem’s enduring difficulty.
• The compilation of 130 references provides a critical, up-to-date resource that maps the evolution of ideas and identifies key open problems in the field.