Construction of nonnegative matrices and the inverse eigenvalue problem

This article presents a technique for combining two matrices, an n x n matrix M and an m x m matrix B, with known spectra to create an (n + m - p) x (n + m - p) matrix N whose spectrum consists of the spectrum of the matrix M and m - p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n + 1 complex numbers (lambda(1), lambda(2), lambda(3), sigma) from a given realizable list of n complex numbers (c(1), c(2), sigma), where c(1) is the Perron eigenvalue, c(2) is a real number and a is a list of n - 2 complex numbers.