The inverse eigenvalue problem for nonnegative matrices (Q703644)

A set of complex numbers is \textit{realizable} if it is the spectrum of an entrywise nonnegative matrix. In [Linear Multilinear Algebra 10, 113--130 (1981; Zbl 0455.15019)] \textit{C. R. Johnson} proved that any realizable set is the spectrum of a entrywise nonnegative matrix with a Perron eigenvector \(e=(1,1,\ldots ,1)^{T}\) (i.e. it is a stochastic matrix). The main result of the paper asserts that if \((\lambda _{1},\lambda _{2},\ldots ,\lambda _{m})\) and \((\mu _{1},\mu _{2},\ldots ,\mu _{m})\) are the spectra of entrywise nonnegative matrices \(A\) and \(B\) respectively, such that \(\mu _{1}\) is a Perron eigenvalue of \(B\) and a maximal diagonal element \(a\) of \(A\) is \(a\geq \mu _{1}\), then the set \((\lambda _{1},\lambda _{2},\ldots ,\lambda _{m},\mu _{2},\ldots ,\mu _{m})\) is realizable and the author construct effectively the matrices. Some interesting methods to obtain new realizable lists from known realizable lists are given in the last section of the paper.