Spectra with positive elementary symmetric functions
Let \(A\) denote an \(n\times n\) matrix with all diagonal entries nonzero and for which the length of the longest cycle in its directed graph is \(k\), \(2\leq k\leq n\). If, in addition, all the cycles in the directed graph of \(-A\) are signed negatively, then the elementary symmetric functions of the eigenvalues of \(A\) are positive and the authors ask whether its eigenvalues lie in the region \(\{z:-\pi+\pi/k<\arg z<\pi-\pi/k\}\). Their approach, in order to investigate the questions that they posed, is to obtain conditions on the location of roots of polynomials in Section 2 and then apply them to the characteristic polynomials of matrices. They prove that it is true for \(k=n-1\) in Section 3. In Section 4, the authors provide a counterexample for the case \(k=n-3\) and discuss related questions for more general classes of matrices with restricted length of the longest cycle.
scientific article; zbMATH DE number 179143
scientific article