The diagonalizable nonnegative inverse eigenvalue problem

In this paper we prove that the SNIEP $\neq$ DNIEP, i.e. the symmetric and diagonalizable nonnegative inverse eigenvalue problems are different. We also show that the minimum $t>0$ for which $(3+t,3-t,-2,-2,-2)$ is realizable by a diagonalizable matrix is $t=1$, and we distinguish diagonalizably realziable lists from general realizable lists using the Jordan Normal Form


Introduction
Classifying spectra of nonnegative matrices is known as the nonnegative inverse eigenvalue problem (or NIEP). The real nonnegative inverse eigenvalue problem (or RNIEP) is to determine necessary and sufficient conditions on the list σ of n real numbers (λ 1 , λ 2 , . . . , λ n ) so that σ is the spectrum of an entry-wise n×n nonnegative matrix A. If there exists such a nonnegative matrix A with spectrum σ, the list σ is said to be realizable or we say the matrix A realizes σ. If we further require that the realizing matrix be symmetric we call the problem the symmetric nonnegative inverse eigenvalue problem (or SNIEP). In a similar vein, the diagonlaizable (real) nonnegative inverse eigenvalue problem (or D-(R)NIEP) requires that the realizing matrix be diagonalizable. If σ is symmetrically realizable, then, since every symmetric matrix is diagonalizable, it follows that σ is D-RNIEP realizable.
Many examples in which realizable lists are not diagonalizably realizable are known [12].
Naturally, one may ask whether every D-RNIEP realizable list is symmetrically realizable. This is the main thrust of this work and an answer in the negative is given.
These problems are unsolved for n ≥ 5 though the trace zero case for 5 × 5 matrices was solved (i) in the case of NIEP by Laffey and Meehan [16] and (ii) in the case of SNIEP by Spector [30]. The RNIEP and SNIEP are the same for n ≤ 4 but are different for n ≥ 5 as was first shown by Johnson, Laffey and Loewy in [9]. However, the NIEP in which we may augment the list σ by adding an arbitrary number N of zeros was solved theoretically by Boyle and Handelman [1] and a constructive version was found by Laffey [14]. For special cases that bound the size of N we refer the reader to [3], [18], [19] The first significant result on NIEP was Suleimanova's result [32] on lists of real numbers with just one positive number, which says that the real list λ 1 > 0 ≥ λ 2 ≥ · · · ≥ λ n is realizable if and only if λ 1 + λ 2 + · · · + λ n ≥ 0. The question of realizing real lists of five or more numbers containing just two positive numbers is still unsolved in general ( [13], [2]).
In this context the spectral radius ρ is known as the Perron root, and for irreducible nonnegative matrices ρ > 0, and it occurs just once as an eigenvalue. We define the Newton power sums s k as follows: Notice that if σ = (λ 1 , λ 2 , . . . , λ n ) is the spectrum of a nonnegative matrix A then the power sum s k is also the trace of the k th power of a realizing matrix A for σ. Independently Loewy and London [23] and Johnson [9] derived an infinite set of inequalities which the spectrum of a nonnegative matrix must satisfy, namely that n m−1 s km ≥ s m k for k, m = 1, 2, . . . known as the JLL conditions. Necessary conditions for realizability in both the RNIEP and the SNIEP thus include: n m−1 s km ≥ s m k for k, m, n = 1, 2, . . . .
A new necessary condition for the SNIEP when n = 5 and when the trace is at least half the spectral radius is given by Loewy and Spector in [31]. A necessary condition for NIEP for general n involving only the first three Newton power sums s k is given by The list σ = {3, 3, −2, −2, −2}, in the guise τ = (1, 1, − 2 3 , − 2 3 , − 2 3 ) was first studied by Salzmann in 1971 [28] and Friedland in 1977 [5]. As can be checked the list τ satisfies the necessary conditions (1), (2) and (3) for all positive integers k, m and n.
It is well known however that the list τ is not realizable. Paparella and Taylor [26] prove that a more general result generalizes lists like τ .
Laffey and Meehan in [17] showed that in order for a list of five numbers which sum to zero to be realizable, a refined JLL inequality must be satisfied, namely 4s 4 − s 2 2 ≥ 0. For σ (which is 3τ ) we have 4s 4 − s 2 2 = 840 − 900 < 0 and so again we see that a small perturbation of σ cannot be realizable.
We define Guo ([7], Theorem 2.1) implies that there is a minimum t > 0 for which However determining the least positive t for which σ t is realizable is not yet solved.
In her thesis, Meehan [25] showed that σ t is realizable for t ≥ 0.519310982048 · · · and a realizing matrix of the form is presented. She also shows that for a 5 × 5 extreme (or Perron extreme) matrix [15], Hence the range for the minimum value of t for which σ t is realizable is 0.39671 · · · ≤ t ≤ 0.51931 · · · but its precise value is not yet determined. This example highlights the difficulty in moving from the NIEP for n = 5 and trace zero to the positive trace case in the NIEP, even when the numbers of the list are all real.
and this is the best possible t in the symmetric case. Thus the RNIEP and the SNIEP are different for n = 5, and this is the first case in which they differ [9]. The interested reader should consult Loewy and London [23], for the proof that the RNIEP=SNIEP for n ≤ 4.

σ
A further related problem of finding the minimum value of t > 0 for which the list is realizable, is solved. Note that the sum of the elements in σ t is zero and so any realizing matrix for σ t must have trace zero. Also note, that any realizing matrix for σ t must be irreducible.
The minimum value of t > 0 for which the symmetric case must be at least one. For a list of five real numbers λ 1 , λ 2 , λ 3 , λ 4 , λ 5 satisfying λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ λ 5 , McDonald and Neumann [24] showed that where A is a symmetric realizing matrix. Hence for σ t we must have that ( and so t = 1 is the best possible result in the symmetric case for σ t . This was first proven by Loewy and Hartwig (unpublished).
However in the general case as noted earlier, Laffey and Meehan show that σ t must satisfy the necessary condition This requires that and they show that the matrix is nonnegative for t ≥ t 0 and has spectrum (3+t, 3−t, −2, −2, −2). Also note, that by the result of Guo [7], the list (3 + t, 3, −2, −2, −2) is realizable for all t ≥ 2t 0 = 0.87598 · · · , but that this is weaker than the bound cited in section 2.1 above.

D-RNIEP = SNIEP
Next we examine the subtle difference between the SNIEP and the Diagonalizable RNIEP or D-RNIEP, where the D-RNIEP is the problem of finding a nonnegative diagonalizable matrix realizing a given real spectrum σ.
However if the list (3 + t, 3 − t, −1.9, −2, −2.1) is to be symmetrically realizable by a matrix A we can use the argument of McDonald and Neumann [24] which yields t ≥ 0.9.
Hence the two problems of D-RNIEP and SNIEP are different at least in this case (see also the recent work on the diagonalizable real nonnegative inverse eigenvalue problem in [10], [11], [12] and [22]).
We also note that for sufficiently small > 0, by continuity, the spectrum (3 + t, 3 − t, −2 + , −2, −2 − ) is diagonalizably realizable (five distinct eigenvalues) for t close to However this is not a continuous property in since we have the following: To prove this result we will make use of the following result of Schur (
Let w T > 0 be the left eigenvector associated with the Perron eigenvalue 3 + t, so that Let v be the right eigenvector associated with the eigenvalue 3 − t, so Av = (3 − t)v.
where the minus sign means the corresponding entry is less than or equal to zero. Hence we can write Av = (3 − t)v as Hence we have that

Now if v has just one positive entry then
Now A diagonalizable also implies there exists a nonsingular matrix T with Now rank(A + 2I) = 2 since rank(A 11 + 2I 2 ) = 2.
Note that A 11 has trace zero and so A 11 + 2I has the form where * and ** do not contribute to the following calculations.
Applying Lemma 2 to . t+ )) . Note that tr(A 22 ) = 0 so upon comparing traces in this equation we get that .
Now (5) gives This implies But note that 24 − 24t > 0 since t < 1, and 24 − 8 t = 16 + 8 − 8 t > 0, implies that the left hand side of this equation is positive and the right hand side is zero which contradicts our hypothesis that t < 1. Hence t ≥ 1.
Laffey and Smigoc [20] proved that (3 + t, 3 − t, −2, −2, −2, 0) is symmetrically realizable by a 6 × 6 matrix for t ≥ 1 3 and we conjecture that t = 1 3 is the best bound for any number of zeros added to the spectrum σ t . Hence the D-RNIEP is different to the general RNIEP. Note that this matrix was built up from a 4 × 4 nonnegative matrix with spectrum (3 + 3 4 , 3 − 3 4 , −2, −2) having the entry 2 on the diagonal using theŠmigoc methods deployed in [29]. The question of whether every realizable spectrum can be realized by a nonderogatory matrix is open and will require further ideas related to those developed in this paper.

Conclusion
In this article we proved that the SNIEP = D-RNIEP and that the D-NIEP can be distinguished from the general NIEP by examining the Jordan Normal Form. To prove the main result (Proposition 1) we used a result of Schur (Lemma 2) and a necessary condition due to McDonald and Neumann (λ 2 + λ 5 ≤ trA) and a result of Courant-