Characterization of the eigenvalues of non-negative matrices

Inverse eigenvalue problems constitute an important subclass of inverse problems that arise in the context of mathematical modelling and parameter identification. The inverse eigenvalue problem for non-negative matrices has a very simple formulation: given a list  = (λ1, λ2, . . . , λn) of complex numbers, find necessary and sufficient conditions for the existence of an n-square non-negative matrix A with spectrum . This problem is a very difficult one and it remains unsolved for any positive integer n. In this work, we will reconstruct the non-negative matrices induced by; Lowey and London for n=3, Reams for n=4, 5, Laffey and Meehan for n=5; by using Newton's identities defined in linear algebra by Dan Kalman. Also, we use Newton's identities to construct the non-negative matrices for n=6,7,8…