Realizable regions for the symmetric nonnegative inverse eigenvalue problem for 6 × 6 matrices (Q6617887)

This manuscript deals with the symmetric nonnegative inverse eigenvalue problem, that is, what are the necessary and sufficient conditions for a multiset of complex numbers \(\{ \lambda_1, \lambda_2, \ldots, \lambda_n \}\) to be the spectrum of a nonnegative symmetric \(n \times n\) matrix? \N\NThis problem is solved only for matrices of size \(n \times n\), \(n \leq 4\). In this paper, the authors work with symmetric nonnegative \(6 \times 6\) matrices. They bound the potential spectra by assuming that the spectral radius is 1; the eigenvalues are ordered so that\N\[\N1 \geq \lambda_2 \geq \cdots \geq \lambda_6 \geq -1,\N\]\Nand the trace is nonnegative, to create the trace nonnegative polytope, \(\mathcal{T}_6\). The Soules method is used to construct symmetric nonnegative matrices with prescribed spectra for most of the trace nonnegative polytope.\N\NThe points in \(\mathcal{T}_6\) that correspond to the eigenvalues of a nonnegative symmetric matrix are called realizable and refer to their collection as the realizable region, \(\mathcal{R}_6\). The authors identify three corners of the trace nonnegative polytope \(\mathcal{T}_6\), dividing the set into points that are realizable, and point that satisfy all known necessary conditions, but their realizability remains unknown. \N\NOne of the main results of this paper establishes the following: The points in \(\mathcal{T}_6\) that satisfy \(1+\lambda_2+\lambda_5+\lambda_6 \geq 0\) and either \(1+\lambda_3+\lambda_4+\lambda_5 \geq 0\), or both \(1+\lambda_4+\lambda_5 \geq 0\) and \(1+\lambda_3+\lambda_6 \geq 0\), are realizable.