The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs

Abstract

The nonnegative inverse eigenvalue problem (NIEP) is: given a family of complex numbers σ = { λ 1 , … , λ n } , find necessary and sufficient conditions for the existence of a nonnegative matrix A of order n with spectrum σ . Loewy and London [R. Loewy, D. London, A note on the inverse eigenvalue problems for nonnegative matrices, Linear and Multilinear Algebra 6 (1978) 83–90] resolved it for n = 3 , and for n = 4 when the spectrum is real. In our way of handling the NIEP, we focus our attention on the coefficients of the characteristic polynomial of A. Thus, the NIEP that we consider is: “ given k 1 , k 2 , … , k n real numbers, find necessary and sufficient conditions for the existence of a nonnegative matrix A of order n with characteristic polynomial x n + k 1 x n - 1 + k 2 x n - 2 + ⋯ + k n ”. The coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix A. We introduce a special type of digraph structure, that we shall call EBL, in which this relation is specially simple. We give some results that show the interest of EBL structures. We completely solve the NIEP from the coefficients of the characteristic polynomial for n = 4 . We also solve a special case of the NIEP for n ⩽ 2 p + 1 with k 1 = ⋯ = k p - 1 = 0 and p ⩾ 2 .