Spectra inhabiting the left half-plane that are universally realizable
A list \(\Lambda :=\left\{ \lambda _{1},\dots,\lambda _{n}\right\} \) of complex numbers is called realizable if it is the spectrum of an \(n\times n\) nonnegative matrix \(A.\) Since a nonnegative matrix has a dominant eigenvalue, we restrict to the case where \(\lambda _{1}\) is real and positive and \(\lambda _{1}\geq \left\vert \lambda _{j}\right\vert \) for all \( j\geq 2\). Moreover, \(\Lambda \) is universally realizable (UR) if in addition \(A\) can be chosen to have any of the Jordan forms allowable by \(\Lambda \). A classical theorem of \textit{H. Minc} [Proc. Am. Math. Soc. 83, 665--669 (1981; Zbl 0472.15006)] shows that if there is a diagonalizable matrix with strictly positive entries which realizes \(\Lambda \) then \(\Lambda \) is UR. Pursuing the efforts which have been made to to relax the conditions in Minc's theorem, the author examines the case where all \(\lambda _{j}\) (except for \(\lambda _{1}\)) lie in the left half-plane and provides new criteria for \(\Lambda \) to be UR. New examples are provided of \(\Lambda \) which are UR but not covered by previous theorems.
scientific article; zbMATH DE number 7476488
scientific article