Speaker
Description
The nonnegative inverse eigenvalue problem (NIEP) seeks to characterize the multisets of complex numbers that can be realized as the spectra of nonnegative matrices. Hessenberg matrices have been used in several central constructive results within this field, including the resolution of the NIEP for $4 \times 4$ matrices, the characterization of spectra where all non-Perron eigenvalues possess negative real parts, and the constructive proof of the Boyle-Handelman theorem.
In this talk, we examine the set of characteristic polynomials realized by nonnegative Hessenberg matrices. To analyze the underlying zero-nonzero patterns of this family of matrices, we introduce an adaptation of the non-symmetric strong spectral property. We apply this modified spectral property to address the following problem: given a prescribed pattern of a nonnegative Hessenberg matrix $A$, we seek to identify perturbations of the characteristic polynomial of $A$ that maintain its realizability with nonnegative Hessenberg matrices.