Construction of nonnegative matrices and the inverse eigenvalue problem
Helena Šmigoc · Linear and Multilinear Algebra · 2005
This article presents a technique for combining two matrices, an n × n matrix M and an m × m matrix B, with known spectra to create an (n + m − p) × (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 (λ1,λ2,λ3,σ) from a given realizable list of n complex numbers (c 1,c 2,σ), where c 1 is the Perron eigenvalue, c 2 is a real number and σ is a list of n − 2 complex numbers.