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The Nonnegative Inverse Eigenvalue Problem: A Theoretical and Computational Investigation
Dissertation

The Nonnegative Inverse Eigenvalue Problem: A Theoretical and Computational Investigation

Benjamin James Clark
Doctor of Philosophy (PhD), Washington State University
2026
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Abstract

inverse eigenvalue problem nonnegative matrix
The Nonnegative Inverse Eigenvalue Problem (NIEP) asks for the necessary and sufficient conditions for a collection of complex numbers to be the spectrum of a nonnegative matrix. While solved for matrices of dimension $n \le 4$ , the problem and its variations remain notoriously open for $n \ge 5$. This dissertation shifts the paradigm from traditional matrix construction techniques to real algebraic geometry. By translating structural matrix constraints into systems of polynomial inequalities, the realizable spectral spaces are analyzed as semialgebraic sets. To systematically analyze these boundaries, a new six-type classification system for the trace polytope is introduced, completely characterizing the realizability of its extreme vertices. Furthermore, two novel numerical algorithms are developed: one to iteratively construct $n$-dimensional feasibility regions and another to efficiently test the exact realizability of targeted candidate spectra. Applying this framework to lower dimensions, this work provides a complete, exact solution for the $4 \times 4$ stochastic symmetric nonnegative inverse eigenvalue problem. It rigorously establishes that the polynomial inequality $(1+\lambda_3)(1+\lambda_4) + (\lambda_2 + \lambda_3)(\lambda_2 + \lambda_4) \ge 0$ is both necessary and sufficient for realizability. Finally, the computational algorithms are deployed to map uncharted boundaries of the $5 \times 5$ SNIEP and RNIEP. By exploring previously unknown two-dimensional slices and three-dimensional regions near the boundary corners, this research generates formalized algebraic conjectures that define the exact polynomial surfaces of the realizable space, providing a robust framework for future proofs.

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