Existence and construction of nonnegative matrices with prescribed spectrum

Symmetric nonnegative realization of spectra

“…This result was first used by Perfect [14] in connection with the NIEP, and has given rise lately to a number of realizability criteria [19,20,22]. Closer to our approach in this paper is Rado's 1 extension (Theorem 2.3 below) of Brauer's result, which was used by 2.…”

“…Kellogg's criterion [8], for instance, was shown by Fiedler [4] to imply symmetric realizability. Radwan [16] proved that Borobia's criterion [1] is also a criterion for symmetric realizability , and Soto's criterion for the RNIEP [19] (Theorem 4.2 below), which contains both Kellogg's and Borobia's criteria [20], was shown in [23] to be also a symmetric realizability criterion. In this section we compare the new result in this paper (Theorem 3.1) with some previous realizability criteria for SNIEP.…”

“…The complex cases n = 4 and n = 5 have been solved for matrices of trace zero by Reams [17] and Laffey and Meehan [10], respectively. Sufficient conditions or realizability criteria for the existence of a nonnegative matrix with a given real spectrum have been obtained in [25,14,15,18,8,1,19,22] (see [3, §2.1] and references therein for a comprehensive survey). If we additionally require the realizing matrix to be symmetric, we have the symmetric nonnegative inverse eigenvalue problem (hereafter SNIEP).…”

“…It is a consequence of Theorem 2.6 in the same way as Theorem 2.4 follows from Theorem 2.3. In section 4 we show that Soto's realizability criterion [19,Theorem 17], which is also sufficient for the symmetric case, is contained in the criterion of Theorem 3.1 below. Example 5.2 in section 5 shows that the inclusion is strict.…”

On nonnegative realization of partitioned spectra

“…, λ n } can be partitioned as Λ = Λ 1 ∪ · · · ∪ Λ r in such a way that each Λ i is realizable by a nonnegative matrix A i , then Λ is realizable by the nonnegative block diagonal matrix A = ⊕ i A i . In [2,16] the authors give conditions under which Λ is realizable even if some of the Λ i 's are not realizable, provided there are other sublists Λ j which are realizable and, in a certain way, compensate the nonrealizability of the former ones. This is done by employing a very useful result, due to Brauer [4], together with the properties of real matrices with constant row sums.…”