The NIEP for four dimensional Leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial
Let \(\Omega_n^0\) denote the set of all points in the complex plane that are eigenvalues of \(n\)-square doubly stochastic matrices with zero trace, and let \(\Pi_k^0\) be the union of the set \(\{ 1 \}\) and the convex hull of all \(k\)-th roots of unity not equal to \(1\). The author [Electron. J. Linear Algebra 30, 599--604 (2015; Zbl 1326.15022)] showed that the following inclusion holds: \[ \Omega_n^0 \supseteq \Pi_2^0 \cup \Pi_3^0 \cup \ldots \cup \Pi_{n-2}^0 \cup [-1,1] \cup \Pi_n^0.\eqno{(1)} \] Now, in the paper under review, the same author proves that, if \(n = 4,\) the equality sign holds in \((1)\). This is achieved by auxiliary results on the trace of powers and products of permutation matrices with zero trace, and by establishing constraints for the coefficients of the characteristic polynomial of an arbitrary 4-square doubly stochastic matrix with zero trace. Moreover, explicit counterexamples are given showing that equality cannot occur in \((1)\) if \(n \in \{ 5, 6 \}\). In the final part of the paper, a full characterization of the set of eigenvalues is obtained in the case of 4-square Leslie stochastic matrices with zero trace. (These consist of arbitrary entries \(a_{1j}, \, 2 \leq j \leq 4,\) such that \(a_{12}+a_{13}+a_{14} = 1, \; a_{21} = a_{32} = a_{43} = 1,\) and \(a_{ij} = 0\) otherwise.)
scientific article; zbMATH DE number 6883439
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