On nonnegative realization of partitioned spectra
Ricardo L. Soto, Óscar Rojo, Cristina Manzaneda · Electronic Journal of Linear Algebra · 2011
Abstract. We consider partitioned lists of real numbers Λ = {λ1, λ2,..., λn}, and give efficient and constructive sufficient conditions for the existence of nonnegative and symmetric nonnegative matrices with spectrum Λ. Our results extend the ones given in [R.L. Soto and O. Rojo. Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl., 416:844– 856, 2006.] and [R.L. Soto, O. Rojo, J. Moro, and A. Borobia. Symmetric nonnegative realization of spectra. Electron. J. Linear Algebra, 16:1–18, 2007.] for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an r × r nonnegative matrix with prescribed complex eigenvalues and diagonal entries. Key words. Nonnegative inverse eigenvalue problem. AMS subject classifications. 15A18. 1. Introduction. The nonnegative inverse eigenvalue problem (NIEP) is the problem of characterizing all possible spectra of entrywise nonnegative matrices. This problem remains unsolved. A complete solution is known only for n ≤ 4 [9, 11, 22]. Sufficient conditions for the existence of a nonnegative matrix with prescribed complex spectrum have been obtained in [2, 13, 14], and recently in [20]. Necessary conditions