The authors present procedures for constructing new realizable lists from old in the nonnegative inverse eigenvalue problem (NIEP).
Given a list of complex numbers \(\sigma :=(\lambda_1, \lambda_2, \dots, \lambda_n)\) they say that \(\sigma\) is realizable if it is the spectrum of some nonnegative matrix \(A\) and in this case, they say that \(A\) realizes \(\sigma\). The NIEP is the problem of characterizing all realizable lists.
In the introduction of this work, the authors present a wide state of the art of this problem. In Section 2, given a realizable list \((\rho, \lambda_2,\lambda_3, \dots, \lambda_m)\) of a matrix \(A\), where \(\rho\) is the Perron eigenvalue, that is the spectral radius of \(A\), and \(\lambda_2\) is real, the authors find families of lists \((\mu_1, \mu_2, \dots, \mu_n)\), for which \[ (\mu_1, \mu_2, \dots, \mu_n, \lambda_3, \dots, \lambda_m) \] is realizable.
Finally, in Section 3, given a realizable list of a matrix \(A\) \[ (\rho, \alpha+i\beta, \alpha-i\beta, \lambda_4, \lambda_5, \dots, \lambda_m), \] where \(\rho\) is the spectral radius of \(A\) and \(\alpha\) and \(\beta\) are real, the authors obtain families of lists \((\mu_1, \mu_2, \mu_3, \mu_4)\), for which \((\mu_1, \mu_2, \mu_3, \mu_4, \lambda_4, \dots, \lambda_m)\) is realizable, where \(\mu_1\), \(\mu_2\), \(\mu_3\) and \(\mu_4\) are the roots of the polynomial \[ q(x)=(x- \rho)((x-\alpha)^2+\beta^2)(x-a)-t((x-\alpha)((1+\eta)x-\alpha-\eta \rho)+\beta^2), \] \(a\), \(t\) and \(\eta\) real numbers satisfying \(a,t \geq 0\) and \(0< \eta \leq 1\).