A multi-set \(\Lambda = \{\lambda_1, \lambda_2,\dots, \lambda_n\}\) of complex numbers is said to be realizable if it is the spectrum of a (entrywise) nonnegative matrix A. The problem of finding necessary and sufficient conditions for \(\Lambda\) being realizable is known as the nonnegative inverse eigenvalue problem (NIEP). It is known as the real nonnegative inverse eigenvalue problem (RNIEP) if all \(\lambda\)’s are required to be real and as the symmetric nonnegative inverse eigenvalue problem (SNIEP) if \(A\) is required real. These problems have been extensively studied in the literature.
\(\Lambda\) is universally realizable (UR) if \(\Lambda\) is realizable for each possible Jordan canonical form allowed by \(\Lambda\), a notion due to C. R. Johnson et al. [Linear Algebra Appl. 587, 302–313 (2020; Zbl 1475.15012)]. Obviously, this imposes more restrictions. For example, in the case of complex conjugate eigenvalues, this means that any Jordan block must occur in conjugate pairs.
The authors give a general realizability criterion, based on the use of normal ODP matrices, which are nonnegative matrices with positive off-diagonal entries. Such criterion gives rise to another specific criterion for a real \(\Lambda\). An additional realizability criterion is obtained. In particular, they show that Soto-\(p\) realizable implies UR and some realizability criteria for RNIEP and SNIEP are realizability criteria for URP as well. Examples are given to illustrate the results.