On the Universal Realizability Problem: New Results

Let Λ=λ1…λn be a list of complex numbers. Λ is said to be realizable if there is a nonnegative matrix with spectrum Λ. The list Λ is said to be universally realizable UR if it is realizable for each possible Jordan canonical form JCF allowed by Λ. The problem of determining the universal realizability of Λ is called universal realizability problem URP. The first results concerning URP (formerly called nonnegative inverse elementary divisors problem) are due to H. Minc and they establish that if Λ is the spectrum of a diagonalizable positive matrix, then Λ is UR. In this chapter, we introduce new results that contain extensions of Minc’s results and that allow us to show the universal realizability of lists of complex numbers not positively realizable. We also prove new universal realizability criteria and structured universal realizability criteria.