The symmetric nonnegative inverse eigenvalue problem for \(5 \times 5\) matrices
Motivated by earlier works of \textit{L. Elsner}, \textit{R. Nabben} and \textit{M. Neumann} [Linear Algebra Appl. 271, 323-343 (1998; Zbl 0891.15019)] and \textit{J.J. McDonald} and \textit{M. Neumann} [Contemp. Math. 259, 387-407 (2000; Zbl 0965.15009)] this paper considers mainly the inverse eigenvalue problem for nonnegative symmetric matrices of order \(5\). The authors deal with the problem of finding the realizable points in the convex hull of \((1,1,1,-1,-1)\), \((1,1,0,-1,-1)\), \((1,0,0,0,-1)\), \((1,1/2,1/2,-1,-1)\), \((1,0,0,-1/2,-1/2)\), denoted by \(\mathcal{U}\). For that purpose they consider the subset \(\mathcal{U}_1\) obtained from \(\mathcal{U}\) by removing a union of two faces of \(\mathcal{U}\), one being of the Soules set and the other consisting of the trace \(0\) matrices. They show that certain matrices can not be principal submatrices of symmetric nonnegative matrices with spectrum in \(\mathcal{U}_1\). The nine \((+,0)\) patterns for which there exists an extreme matrix with spectrum in \(\mathcal{U}_1\) are analyzed.
symmetric nonnegative inverse eigenvalue problem
scientific article; zbMATH DE number 2126356