Perron similarities and the nonnegative inverse eigenvalue problem

[2409.07682] Perron similarities and the nonnegative inverse eigenvalue problem Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Spectral Theory arXiv:2409.07682 (math) [Submitted on 12 Sep 2024 ( v1 ), last revised 31 Jul 2025 (this version, v4)] Title: Perron similarities and the nonnegative inverse eigenvalue problem Authors: Charles R. Johnson , Pietro Paparella View a PDF of the paper titled Perron similarities and the nonnegative inverse eigenvalue problem, by Charles R. Johnson and Pietro Paparella View PDF HTML (experimental) Abstract: The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevich arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of $4 \times 4$ stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices. Comments: To appear in the Trans. Amer. Math. Soc Subjects: Spectral Theory (math.SP) ; Rings and Algebras (math.RA) MSC classes: Primary: 15A29, Secondary: 15B48, 15B51 Cite as: arXiv:2409.07682 [math.SP] (or arXiv:2409.07682v4 [math.SP] for this version) https://doi.org/10.48550/arXiv.2409.07682 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Trans. Amer. Math. Soc., 378: 8361--8389, 2025 Related DOI : https://doi.org/10.1090/tran/9596 Focus to learn more DOI(s) linking to related resources Submission history From: Pietro Paparella [ view email ] [v1] Thu, 12 Sep 2024 01:11:00 UTC (348 KB) [v2] Tue, 24 Sep 2024 21:55:12 UTC (348 KB) [v3] Sat, 5 Oct 2024 19:24:37 UTC (395 KB) [v4] Thu, 31 Jul 2025 22:10:58 UTC (395 KB) Full-text links: Access Paper: View a PDF of the paper titled Perron similarities and the nonnegative inverse eigenvalue problem, by Charles R. Johnson and Pietro Paparella View PDF HTML (experimental) TeX Source view license Current browse context: math.SP < prev | next > new | recent | 2024-09 Change to browse by: math math.RA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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