On new universal realizability criteria

: A list λ λ λ Λ , , , n 1 2 { } = … of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix and is said to be universally realizable ( UR ) , if it is realizable for each possible Jordan canonical form allowed by Λ . In 1981, Minc proved that if Λ is diagonalizably positively realizable, then Λ is UR [ Proc. Amer. Math. Society 83 ( 1981 ) , 665 – 669 ] . The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins ’ s result have been obtained by Soto et al. [ Spec. Matrices 6 ( 2018 ) , 301 – 309 ] , [ Linear Algebra Appl. 587 ( 2020 ) , 302 – 313 ] . In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ is UR, then for t 0 ≥ ,


Introduction
A list λ λ λ Λ , , , n of complex numbers is said to be realizable if it is the spectrum of an n-by-n entrywise nonnegative matrix (called the realizing matrix), with λ 1 being the Perron eigenvalue.Λ is diagonalizably realizable (DR) if the realizing matrix is diagonalizable, and Λ is universally realizable (UR) if it is realizable for each possible Jordan canonical form (JCF) allowed by Λ.The problems of realiz- ability and universal realizability of spectra are called nonnegative inverse eigenvalue problem (NIEP) and universal realizablity problem (URP), respectively.The URP contains the NIEP, and both problems are equivalent if the given complex numbers are distinct.Both problems have been solved only for lists of n 4 ≤ complex numbers, which shows the difficulty of them.A complete solution is still far from the present state of the art about these problems.
The first known works on the URP (formerly called the nonnegative inverse elementary divisors problem) are due to Minc [1].In particular, Minc showed that if a list λ λ λ Λ , , , n of complex numbers is positively DR, then Λ is UR.The positivity and diagonalizability conditions are necessary for Minc's proof, and the question set by Minc himself, whether the result holds for nonnegative realizations, has been open for almost 40 years.Although nonpositive universal realizations are known for certain spectra on the left half-plane (see [2][3][4]), only recently have two extensions of the Minc's result been obtained in [5,6].In [5], Collao et al. proved that if Λ is DR by a matrix with constant row sums and a positive column, then Λ is UR.In [6], Johnson et al. introduced the concept of off-diagonal positive (ODP) matrices, that is, nonnegative matrices with all their off-diagonal entries being positive (zero diagonal entries are permitted) and proved that if Λ is diagonalizably ODP realizable, then Λ is also UR.Note that the set of matrices satisfying both extensions contains the set of positive matrices.Then, these extensions allow us to significantly increase the set of spectra that can be shown to be UR.Moreover, from [6], certain trace zero spectra can be shown to be UR, which is not possible from the Minc's result.
Regarding nonpositive universal realizations, we mention the lists of complex numbers λ λ λ Λ , , , n { } = … , with λ λ λ λ i n 0, Re 0, Re Im , 2, 3, , , lists of Suleimanova type [7], and λ λ λ λ i n 0, Re 0, 3 Re Im , 2, 3, , , lists of Šmigoc type [8].The lists in (1) and (2) were proved, respectively in [3] and [4], to be UR if and only if they are realizable if and only if λ 0 A set K of conditions is said to be a realizability criterion, if any list Λ satisfying conditions K is realizable.In this case, we say that Λ is K -realizable.Following [9], we define the K -negativity of a list λ λ λ Λ , , , n and we define the K -realizability margin of Λ as is said to have constant row sums, if all its rows sum a same constant α.We denote by α the set of all n-by-n real matrices with constant row sums equal to α.It is clear that any matrix in α has an eigenvector e 1, ,1 T [ ] = … corresponding to the eigenvalue α.Let e k be the vector with 1 in position k and zeros elsewhere.The relevance of the real matrices with constant row sums is due to the fact that the problem of finding a nonnegative matrix with spectrum λ λ λ Λ , , , n is equivalent to the problem of finding a nonnegative matrix in λ1 with spectrum Λ [10].In [11], Soto introduced two realizability criteria, S1 and S2, for the RNIEP to have a solution, that is, the NIEP with λ λ λ Λ , , , n having only real numbers.In [12,13], it was later shown that S1 and S2 are also symmetric realizability criteria, that is, with the realizing matrix being symmetric.In [14], a new realizability criterion, S3, was introduced, which is easily adaptable to be a symmetric realizability criterion as well.In a natural way, S3 criterion extends to a family of Soto-p realizability criteria for both problems, the real and symmetric NIEP to have a solution [14].The importance of Soto-p criteria was established by Ellard and Šmigoc in [15].Therein the authors prove that the following four symmetric realizability criteria are equivalent and contain all previously known symmetric realizability criteria in the literature (except the criterion in [16]): criterion by Ellard and Šmigoc [15], Soules criterion [17], C-realizability criterion by Borobia et al. [18], and Soto-p family criteria by Soto [14].In [19], the authors of real numbers is Soto-p realizable, then Λ is diagonalizably ODP realizable, and therefore, it is UR.Recently in [20], S1 and S2 criteria were extended to spectra of complex numbers as CS1, CS2, and E-CS2 criteria, respectively.In this work, we show how to construct, from CS1, CS2, and E-CS2 criteria, diagonalizable realizing matrices with a given spectrum Λ for each possible JCF allowed by Λ. Next, we recall these criteria with the following theorems: { } = … be a list of complex numbers, where λ λ λ p > ≥⋯≥ are real and λ λ λ , , , where S λ λ ( ) ( ) = be the CS1-negativity of the list Λ k , k t 2, , = … and the CS1-realizability margin of Λ , 1 respectively.Let Λ is realizable.
To apply criterion CS2, we measure, from CS1 criterion, the negativity of the lists Λ k .Since the nega- tivity of a list depends on the criterion used, then the negativity of Λ k may be measured by applying a more efficient criterion, which in many cases may give us a better information about the realizability of Λ k .This is precisely what the criterion E-CS2 does.In particular, criterion E-CS2 extends criterion CS2 by taking a convenient criterion K to measure the negativity of Λ k , k t 2, , = … and the realizability margin of Λ 1 .Then, (4) becomes with In [21], in the case of reducible realizations, the authors showed that the union of two UR lists is not necessarily UR.Here, we prove that under certain conditions, if Λ 1 and Λ 2 are DR, then Λ Λ 1 2 ∪ becomes positively DR and therefore UR.
Guo [22] proved that if In what follows, we will use the following perturbation result by Rado, published by Perfect [23], which shows how to change r eigenvalues of an n-by-n matrix, without changing any of the remaining n r − eigenvalues.
Theorem 1.3.[23] Let A be an n-by-n arbitrary matrix with spectrum λ λ λ Λ , , , n The case r 1 = is the well-known Brauer's theorem.In [20], the authors prove a diagonalizable version of the Rado's perturbation result, Theorem 1.3, which will also be used to obtain some of our results.
Theorem 1.4.[20] Let A be an n-by-n diagonalizable matrix with spectrum This article is organized as follows: In Section 2, we show how to construct a diagonalizable ODPrealizing matrix with a prescribed complex spectrum.Moreover, we also show that the realizable lists in (2), are, in particular, the spectrum of a diagonalizable ODP matrix.In Section 3, we show how to construct, from CS1, CS2, and E-CS2 criteria, realizing matrices with a given spectrum Λ for each JCF allowed by Λ.In Section 4, we show that if Λ 1 and Λ 2 are UR lists of complex numbers, then under certain conditions Λ Λ 1 2 ∪ is also UR.In this section, we also show that the CS1 universal realizability criterion satisfies the Guo perturbations (see [22]), that is, if We also introduce examples to illustrate the results.

ODP-realizing matrices
In this section, we show how to construct an ODP matrix A with prescribed complex spectrum λ λ λ Λ , , , n { } = … be a realizable list of complex numbers.If we may decompose Λ as , , where some lists Λ k can be empty, such that , and (ii) there is a p 0 -by-p 0 diagonalizable ODP matrix with spectrum Λ 0 and diagonal entries ω ω ω , , , p … is diagonalizably ODP realizable.In this case, to decide the universal realizability of Λ is not necessary to compute a nonnegative matrix with spectrum Λ for each JCF associated with Λ.It is clear that a list in (1) (list of complex Suleimanova type), with λ λ Re Im Proof.Without loss of generality, we consider the case λ 0 The condition is necessary.For the sufficiency, we take and the auxiliary lists where the lists Λ k 2 are of the form a bi a bi Λ , Finally, we apply Theorem 1.4 to obtain a diagonalizable ODP matrix with spectrum Λ.Hence, Λ is UR.□ 3 The universal realizability of CS1, CS2, and E-CS2 criteria That is, a list Λ not Perron extreme with Λ ε DR is UR.This fact can be applied to criteria CS1, CS2, and E-CS2, to show that they become universal realizability criteria if we take strict inequality in the corresponding conditions (3)-( 5).However, this is not possible if the diagonalizability condition of Λ ε is not satisfied.There are several examples of realizable lists CS1, CS2, and E-CS2 that are not DR.This is the reason why we place the emphasis on constructing a diagonalizable realizing matrix for Λ.The construc- tion that we propose makes it possible for an initially nondiagonalizably CS1, CS2, or E-CS2 realizable list to become so.For instance, the list i i Λ .
However, A is not diagonalizable, and therefore, Λ is not CS1 UR (observe that Λ is Rado diagonalizable).If we take ε 14 + as the Perron eigenvalue in Λ, the new realizing matrix (with the same CS1 procedure) will also be not diagonalizable.Then, we may take A B eq T = + with B 5 0 which is DR.Now, we study the universal realizability of criteria CS1,CS2, and E-CS2.Our strategy is to partition the given list into sub-lists, Λ Λ Λ 1 2 = ∪ , with Λ 1 and Λ 2 containing the real and complex numbers, respectively, in the case of criterion CS1.Next, we merge Λ 1 and Λ 2 to obtain a diagonalizable realizing matrix in CS λ1 with a positive column and spectrum Λ.Then, from the extension in [5], Λ is UR.For CS2 criterion, we take = ∪ ∪⋯∪ , where some lists may be CS1 realizable, while the others are not.We first transform each sub-list Λ t into a diagonalizably positively realizable list.Then, we merge them in a 2-by-2 way and apply Theorem 1.4 to obtain a diagonalizable positive realizing matrix for Λ.
then Λ is UR.
Proof.Let be the right side in (6).Then, there is a real number ε 0 > such that λ ε 1 = + .Consider the partition is S1 realizable, and therefore, it is also symmetrically (diagonalizably) realizable.Then, from Fiedler's result [26], is positively symmetrically realizable.Let B 0 > be the realizing matrix for Γ .
Since the matrices C kk are normal, they are diagonalizable.Thus, the block diagonal matrix C is also diagonalizable.Then, for an n p p ( ) − × matrix F with appropriate positive entries on its first column and zeros elsewhere, we have that is diagonalizable nonnegative in CS λ1 with spectrum Λ and with its first column being positive, where q q q q , , , Then, from the extension in [5], Λ is UR.
Proof.Remember that with Λ 1 being CS1 realizable.Then, is positively UR with a diagonalizable positive realizing matrix A .
Now, some lists Λ k can be CS1 realizable, while others are not.Suppose that Λ k , k r 2, , = … not CS1 realizable, while Λ j , j r t 1, , = + … , are.Then, for appropriate small values δ 0, k1 > the lists Λ k that are not CS1 realizable become which are now also UR and without loss of generality we assume that they have a diagonalizable positive realizing matrix , which are UR with a diagonalizable positive realizing matrix Next, in the first step of the procedure, we apply Theorem 1.4 to the matrices A 1 and A 2 to obtain which will be diagonalizable positive with spectrum ( ) exists, and we obtain the matrix M 1 in (8).
In the second step of the procedure, we apply Theorem 1.4 to the matrices M 1 and A .
is diagonalizable positive, and we obtain a diagonalizable positive matrix M 2 with spectrum The procedure follows similar steps for the matrices M k and For the matrices A j , j r t 1, , = + … , we do the same, except that in this case, we have Λ 0.
CS j 1 ( ) = Then, the eigenvalues and diagonal entries in the 2-by-2 matrix B j are μ λ , + + , respectively, and we have In the last step, (t 1 − )th step, we take the matrices M t 2 − and A t to obtain M t 1 − with spectrum Λ. Observe that ( ) = = , and λ 12. M = Then (4) is satisfied.We will show that Λ is UR.Since Λ 1 is CS1 realizable, we obtain for i with Proof.The proof is similar to the proof of Theorem 3.2, but in this case, we must take care that the K-criterion that we apply to each sub-list where SR and LL are the criteria in [27, Lemma 1] and [24], respectively, then Λ is realizable.For = .Then, where q 0 0 0 0 0 0 0 0 0

On the union of UR lists
In [21], authors showed that, in general, the union of two UR lists is not necessarily UR.In this section, we prove that under certain conditions, if Λ 1 and Λ 2 are DR, then Λ Λ such matrix exists.Then, the matrix Observe that in Theorem 4.1, diagonalizable realizability is enough to prove universal realizability if (10) is satisfied.The extension in [5,6] allows us to relax the conditions to prove that the union of two UR lists is also UR.Then, we have the following corollaries.
Proof.Let A and B be the realizing matrices for Λ 1 and Λ 2 , respectively.We assume, without loss of generality, that the first column of A is positive and that A α1 ∈ and B .
∪ and with its first column being positive.Hence, Λ Λ Guo [22] proved that increasing by t the Perron eigenvalue and decreasing or increasing by t, t 0, > another real eigenvalue of a realizable list of complex numbers preserves the realizability.In [2] is also UR.In [21], the authors show that this is not true for general lists.In this article, we show that a CS1 UR list of complex numbers satisfies the Guo perturbations [22].It is clear that the block diagonal matrix C is also diagonalizable.Then, for an n p p ( ) − × matrix F with appropriate positive entries on its first column and zeros elsewhere, we have that and with its first column being positive, where q q q q , , , … with q i p q λ λ k p n 0, 1, 2, , .min 0, Re , Im , 1, , .Then, from the extension in [5], Λ t is CS1 UR. □

ε 1 ,
Then, we may take the diagonalizable positive matrix B CS ,

=
Now, we apply Theorem 1.4 to the matrices A 1 and A 2 .First, we need a matrix B with eigenvalues 14 and 6 and diagonal entries25   2 from the necessary and sufficient condition in [27, Lemma 1], the realizing symmetric circulant matrix

F 4
spectrum Λ and with at least one positive column.Hence, Λ is UR.

Theorem
let B be a diagonalizable ODP matrix with spectrum Λ 0 and diagonal entries ω ω ω , , , p Γ 11 and Γ 12 are diagonalizably ODP realizable by there exists a diagonalizable positive matrix B CS β ε Now, we apply Theorem 1.4 to matrices A and B. First, we need to compute a 2-by-2 diagonalizable positive matrix with eigenvalues α 1 and β , 1 and diagonal entries g α Λ On new universal realizability criteria  13