The Nonnegative Inverse Eigenvalue Problem
An open-source, interactive collection of computational tools and geometric visualizations designed to explore, understand, and advance research on matrix spectra and nonnegativity.
Interactive Visualizations & Solvers
Select a tool to begin interactive spectral analysis
Karpelevič Region Viewer
Visualize the exact region 𝒦n of all complex numbers that can arise as eigenvalues of an n × n stochastic matrix. Features live Farey sequence root calculation, Ito boundary polynomial solver, point containment testing, and dynamic power orbit visualization (z¹, z², …, zᵏ).
Trace Polytope Gallery
Explore interactive 2D polygons and 3D polyhedra capturing necessary and sufficient trace polytope conditions for the Symmetric (SNIEP) and Real (RNIEP) inverse eigenvalue problems for dimensions n = 4, 5, and 6. Filter by matrix class, dimension, and vertex signature.
About the NIEP
Mathematical foundations and research context
The Core Problem
The Nonnegative Inverse Eigenvalue Problem (NIEP) asks: Given a multiset of n complex numbers σ = {λ₁, λ₂, …, λₙ}, does there exist an n × n entrywise nonnegative matrix A ≥ 0 whose spectrum is σ?
If the matrix is restricted to be symmetric, the question is termed the Symmetric NIEP (SNIEP); if restricted to have real entries, it is the Real NIEP (RNIEP).
Perron-Frobenius & Trace Conditions
Every nonnegative matrix satisfies the fundamental Perron-Frobenius Theorem: the spectral radius ρ(A) is itself an eigenvalue with an associated nonnegative eigenvector.
Furthermore, because all diagonal entries of powers Aᵏ are nonnegative, the matrix traces satisfy tr(Aᵏ) = ∑ λᵢᵏ ≥ 0 for all positive integers k, generating infinite families of polynomial inequalities.
Karpelevič's Milestone (1951)
In 1951, F. I. Karpelevič characterized the set 𝒦n of all possible eigenvalues of n × n stochastic matrices.
The boundary ∂𝒦n consists of curvilinear arcs connecting roots of unity corresponding to Farey fractions, described by algebraic equations developed by Ito (1997) and Johnson & Paparella.