On new universal realizability criteria

Luis E. Arrieta and Ricardo L. Soto
From the journal Special Matrices

Abstract

A list Λ = { λ 1 , λ 2 , , λ n } 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 , Λ t = { λ 1 + t , λ 2 ± t , λ 3 , , λ n } is also UR.

MSC 2010: 15A20; 15A29

1 Introduction

A list Λ = { λ 1 , λ 2 , , 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 realizability 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 1 , λ 2 , , 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 1 , λ 2 , , λ n } , with

(1) λ 1 > 0 , Re λ i 0 , Re λ i Im λ i , i = 2 , 3 , , n ,

lists of Suleimanova type [7], and

(2) λ 1 > 0 , Re λ i 0 , 3 Re λ i Im λ i , i = 2 , 3 , , n ,

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 i = 1 n λ i 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 1 , λ 2 , , λ n } as

N K ( Λ ) = min { δ 0 : { λ 1 + δ , λ 2 , , λ n } is K -realizable } ,

and we define the K -realizability margin of Λ as

K ( Λ ) = max { ε 0 : { λ 1 ε , λ 2 , , λ n } is K -realizable } ,

with λ 1 ε being the Perron eigenvalue of the modified list.

An n -by- n matrix A = ( a i j ) is said to have constant row sums, if all its rows sum a same constant α . We denote by CS α the set of all n -by- n real matrices with constant row sums equal to α . It is clear that any matrix in CS 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 1 , λ 2 , , λ n } is equivalent to the problem of finding a nonnegative matrix in CS λ 1 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 1 , λ 2 , , 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 proved that if a list Λ = { λ 1 , λ 2 , , λ n } 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:

Theorem 1.1

[20] CS1 Let Λ = { λ 1 , λ 2 , , λ n } be a list of complex numbers, where λ 1 > λ 2 λ p are real and λ p + 1 , λ p + 2 , , λ n are conjugate complex. If

(3) λ 1 λ p S k < 0 S k i = p + 1 n min { 0 , Re λ i , Im λ i } + ( ξ λ p ) Only if ξ > λ p ,

where S k = λ k + λ p k + 1 , k = 2 , 3 , , p 2 ; S p + 1 2 = min { λ p + 1 2 , 0 } for p odd; and ξ = max p + 1 i n { Re λ i + Im λ i } , then Λ is realizable.

Theorem 1.2

[20] CS2 Let Λ = { λ 1 , λ 2 , , λ n } be a list of complex numbers, where λ 1 > λ 2 λ p are real, and λ p + 1 , λ p + 2 , , λ n are conjugate complex. Suppose that

Λ = Λ 1 Λ 2 Λ t , Λ k = { λ k 1 , λ k 2 , , λ k s k } , k = 1 , , t , λ 11 = λ 1 , λ k 1 0 ,

with Λ 1 being CS1 realizable.

Let

T k = N C S 1 ( Λ k ) and C S 1 ( Λ 1 )

be the C S 1 -negativity of the list Λ k , k = 2 , , t and the C S 1 -realizability margin of Λ 1 , respectively. Let

λ M = max { λ 1 C S 1 ( Λ 1 ) ; max 2 k t { λ k 1 } } .

Then, if

(4) λ 1 λ M + k = 2 t T k ,

Λ is realizable.

To apply criterion CS2, we measure, from CS1 criterion, the negativity of the lists Λ k . Since the negativity 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 = 2 , , t and the realizability margin of Λ 1 . Then, (4) becomes

(5) λ 1 λ M + k = 2 t N K ( Λ k ) , k = 2 , , t ,

with

λ M = max { λ 1 K ( Λ 1 ) ; max 2 k t { λ k 1 } } .

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 1 Λ 2 becomes positively DR and therefore UR.

Guo [22] proved that if Λ = { λ 1 , λ 2 , , λ n } is a realizable list of complex numbers and λ 2 is real, then the list Λ t = { λ 1 + t , λ 2 ± t , λ 3 , , λ n } , t 0 , is also realizable. In this work, we show that if a list Λ = { λ 1 , λ 2 , , λ n } of complex numbers is CS1 UR, then Λ t = { λ 1 + t , λ 2 ± t , λ 3 , , λ n } is also CS1 UR.

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 1 , λ 2 , , λ n } . Let X = [ x 1 x 2 x r ] be such that rank ( X ) = r and A x i = λ i x i , i = 1 , 2 , , r , r n . Let C be an r × n arbitrary matrix. Then, the matrix A + X C has eigenvalues μ 1 , μ 2 , , μ r , λ r + 1 , λ r + 2 , , λ n , where μ 1 , μ 2 , , μ r are eigenvalues of the matrix Ω + C X with Ω = diag { λ 1 , λ 2 , , λ r } .

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 Λ = { λ 1 , λ 2 , , λ n } . Let X = [ x 1 x 2 x r ] be an n -by- r matrix with rank ( X ) = r , r < n , such that A x i = λ i x i , i = 1 , , r . Let B be an r -by- r diagonalizable matrix with diagonal entries λ 1 , λ 2 , , λ r ; Ω = diag { λ 1 , λ 2 , , λ r } and let C = B Ω . Let J ( A ) = S 1 A S be the J C F of A with S = [ X Y ] , and S 1 = X ˜ Y ˜ . Then, the matrix A + X C X ˜ is diagonalizable with spectrum μ 1 , , μ r , λ r + 1 , , λ n , where μ 1 , , μ r are eigenvalues of B .

This article is organized as follows: In Section 2, we show how to construct a diagonalizable ODP-realizing matrix with a prescribed complex spectrum. Moreover, we also show that the realizable lists in (2), with 3 Re λ i > Im λ i , i = 2 , 3 , , r , 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 1 Λ 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 Λ = { λ 1 , λ 2 , , λ n } is CS1-UR, then Λ t = { λ 1 + t , λ 2 ± t , λ 3 , , λ n } is also CS1-UR. We also introduce examples to illustrate the results.

2 ODP-realizing matrices

In this section, we show how to construct an ODP matrix A with prescribed complex spectrum Λ = { λ 1 , λ 2 , , λ n } . In addition, if A is diagonalizable, then Λ will be UR.

Theorem 2.1

Let Λ = { λ 1 , λ 2 , , λ n } be a realizable list of complex numbers. If we may decompose Λ as

Λ = Λ 0 Λ 1 Λ t with Λ 0 = { λ 01 , λ 02 , , λ 0 p 0 } , λ 01 = λ 1 Λ k = { λ k 1 , λ k 2 , , λ k p k } , k = 1 , 2 , , p 0 ,

where some lists Λ k can be empty, such that

1. for each k = 1 , , p 0 , there is a list Γ k = { ω k } Λ k , 0 ω k λ 1 , diagonalizably ODP realizable by a matrix A k CS ω k , and

2. there is a p 0 -by- p 0 diagonalizable ODP matrix with spectrum Λ 0 and diagonal entries ω 1 , ω 2 , , ω p 0 , then Λ is UR.

Proof

From (i) A = A 1 A 2 A p 0 is diagonalizable nonnegative with spectrum k = 1 p 0 Γ k .

From (ii), let B be a diagonalizable ODP matrix with spectrum Λ 0 and diagonal entries ω 1 , ω 2 , , ω p 0 . Then, we apply Theorem 1.4 (the Rado diagonalizable theorem) to obtain a diagonalizable ODP matrix A + X C X ˜ with spectrum Λ . Therefore, from [6]  Λ , is UR.□

Example 2.1

The list Λ = { 24 , 10 , 10 , 1 ± 10 i , 1 ± 10 i } is out of the Šmigoc region on the left half-plane. We take

Λ 0 = { 24 } , Λ 1 = { 10 , 10 , 1 ± 10 i , 1 ± 10 i } with Γ 1 = { 24 , 10 , 10 , 1 ± 10 i , 1 ± 10 i } and Γ 11 = { 24 , 10 , 1 ± 10 i } , Γ 12 = { 12 , 10 , 1 ± 10 i } .

Then, Γ 11 and Γ 12 are diagonalizably ODP realizable by

A 11 = 0 21 2 1 25 2 1 2 0 21 2 13 1 1 2 0 45 2 21 2 1 1 2 12 and A 12 = 0 21 2 1 1 2 1 2 0 21 2 1 1 1 2 0 21 2 21 2 1 1 2 0 ,

respectively, where A 12 is a circulant matrix and A 11 = A 12 + 12 ee 4 T . From the Šmigoc’s glue technique [8], we obtain the diagonalizable ODP matrix

A = 0 21 2 1 25 8 25 8 25 8 25 8 1 2 0 21 2 13 4 13 4 13 4 13 4 1 1 2 0 45 8 45 8 45 8 45 8 21 2 1 1 2 0 21 2 1 1 2 21 2 1 1 2 1 2 0 21 2 1 21 2 1 1 2 1 1 2 0 21 2 21 2 1 1 2 21 2 1 1 2 0

with spectrum Λ . Observe that from Theorem 2.1, B = [ 24 ] and C = 0 . Then, A + X C X ˜ = A and Λ is UR.

It was proved in [4] that a list in (2) (list of Šmigoc type) is UR if and only if i = 1 n λ i 0 . The following result shows that a list Λ as in (2) with 3 Re λ i > Im λ i , i = 2 , 3 , n , 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 Re λ i > Im λ i , i = 2 , , n , is also diagonalizably ODP realizable.

Corollary 2.1

Let Λ = { λ 1 , λ 2 , , λ n } be a realizable list of complex numbers of Šmigoc type, where λ 1 , , λ p are real and λ p + 1 , , λ n are complex conjugate numbers, with 3 Re λ i > Im λ i , i = p + 1 , n . Then, Λ is UR if and only if i = 1 n λ i 0 .

Proof

Without loss of generality, we consider the case i = 1 n λ i = 0 . The condition is necessary. For the sufficiency, we take Λ = Λ 0 Λ 1 Λ 2 with

Λ 0 = { λ 1 , λ 2 } , Λ 1 = { λ 3 , , λ p } , Λ 2 = { λ p + 1 , , λ n } , i = 1 n λ i = 0 ,

and the auxiliary lists

Γ 1 = i = 3 p λ i , λ 3 , , λ p , Γ 2 = p + 1 n λ i Λ 2 .

Γ 1 is clearly diagonalizably ODP realizable by a ( p 2 ) -by- ( p 2 ) matrix A 1 . To realize Γ 2 , we consider

Γ 2 = p + 1 n λ i Λ 21 Λ 22 Λ 2 r ,

where the lists Λ 2 k are of the form Λ 2 k = { a + b i , a b i } with auxiliary lists Γ 2 k = { γ 2 k , a + b i , a b i } , k = 1 , 2 , r , where

γ 21 = p + 1 n λ i , γ 22 = tr ( Γ 21 ) , γ 23 = tr ( Γ 22 ) , , γ 2 r = tr ( Γ 2 ( r 1 ) ) ,

and tr ( Γ 2 k ) is the sum of elements in Γ 2 k . Observe that from Loewy and London [24] and Brauer [25] results, each list Γ 2 k is diagonalizably ODP realizable by a matrix A 2 k with tr ( Γ 2 k ) in position ( 3 , 3 ). Then, we apply the Šmigoc’s glue technique to the matrices A 2 k to obtain a ( n p ) -by- ( n p ) diagonalizable ODP matrix A 2 with spectrum Γ 2 .

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

A realizable list Λ = { λ 1 , λ 2 , , λ n } of complex numbers is said to be Perron extreme, if the list Λ ε = { λ 1 ε , λ 2 , , λ n } is not realizable for every ε > 0 . If Λ is not Perron extreme, there is an ε > 0 such that Λ ε is realizable. Then, if Λ ε is realizable by a diagonalizable matrix B CS λ 1 ε , A = B + eq T with q T = ε n , , ε n is diagonalizable positive with spectrum Λ . Hence, Λ is UR. 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 construction that we propose makes it possible for an initially nondiagonalizably CS1, CS2, or E-CS2 realizable list to become so. For instance,

the list Λ = { 14 , 5 , 5 , 1 + 3 i , 1 3 i } is CS1 realizable by

A = 0 0 0 0 0 5 5 0 0 0 0 5 5 0 0 0 4 0 1 3 0 2 0 3 1 + 1 1 1 1 1 [ 0 , 5 , 5 , 1 , 3 ] = 0 5 5 1 3 5 0 5 1 3 0 10 0 1 3 0 9 5 0 0 0 3 5 4 2 .

However, A is not diagonalizable, and therefore, Λ is not CS1 UR (observe that Λ is Rado diagonalizable). If we take 14 + ε 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 = 0 0 0 0 0 5 5 0 0 0 5 0 5 0 0 0 4 0 1 3 0 2 0 3 1 ,

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 1 Λ 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 C S λ 1 with a positive column and spectrum Λ . Then, from the extension in [5], Λ is UR. For CS2 criterion, we take Λ = Λ 1 Λ 2 Λ t , 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 .

Theorem 3.1

Let Λ = { λ 1 , λ 2 , , λ n } be aCS1 realizable list of complex numbers with λ 1 > λ 2 λ p being real and λ p + 1 , λ p + 2 , , λ n being conjugate complex. If

(6) λ 1 > λ p S k < 0 S k i = p + 1 n min { 0 , Re λ i , Im λ i } + ( ξ λ p ) Only if ξ > λ p ,

then Λ is UR.

Proof

Let be the right side in (6). Then, there is a real number ε > 0 such that λ 1 = + ε . Consider the partition

Λ = Λ 1 Λ 2 with Λ 1 = { λ 1 , , λ p } Λ 2 = { λ p + 1 , , λ n } .

It is clear that the list

Γ 1 = λ p S k < 0 S k , λ 2 , , λ p

is S1 realizable, and therefore, it is also symmetrically (diagonalizably) realizable. Then, from Fiedler’s result [26],

Γ 1 , ε = λ p S k < 0 S k + ( ξ λ p ) + ε Only if ξ > λ p , λ 2 , , λ p

is positively symmetrically realizable. Let B > 0 be the realizing matrix for Γ 1 , ε . Then, we may take the diagonalizable positive matrix B C S β , which is similar to B , where β = λ p S k < 0 S k + ( ξ λ p ) + ε Only if ξ > λ p .

Now, we consider the list Λ 2 of complex numbers. Let

C = C p + 1 , p + 1 C n 1 , n 1 with C k k = Re λ k Im λ k Im λ k Re λ k , k = p + 2 j + 1 , j = 0 , 1 , , n 2 p 2 .

Since the matrices C k k 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

B 0 F C C S β and A = B 0 F C + eq T

is diagonalizable nonnegative in C S λ 1 with spectrum Λ and with its first column being positive, where q T = [ q 1 , q 2 , , q n ] , with

q i = 0 , i = 1 , 2 , , p . q k = min { 0 , Re λ k , Im λ k } , k = p + 1 , , n .

Then, from the extension in [5], Λ is UR.□

Theorem 3.2

Let Λ = { λ 1 , λ 2 , , λ n } be aCS2 realizable list of complex numbers with λ 1 > λ 2 λ p being real and λ p + 1 , λ p + 2 , , λ n being conjugate complex. If

(7) λ 1 > λ M + k = 2 t T k ,

where T k = N C S 1 ( Λ k ) and

λ M = max { λ 1 C S 1 ( Λ 1 ) ; max 2 k t { λ k 1 } } ,

then Λ is UR.

Proof

Remember that

Λ = Λ 1 Λ 2 Λ t with Λ k = { λ k 1 , λ k 2 , , λ k s k } , k = 1 , , t , λ 11 = λ 1 , λ k 1 0 ,

with Λ 1 being CS1 realizable. Then,

Λ 1 = { λ M , λ 12 , , λ 1 p 1 }

is also CS1 realizable and Γ 1 = { λ M + ε 1 , λ 12 , , λ 1 p 1 } is positively UR with a diagonalizable positive realizing matrix A 1 CS λ M + ε 1 . Now, some lists Λ k can be CS1 realizable, while others are not. Suppose that Λ k , k = 2 , r , are not CS1 realizable, while Λ j , j = r + 1 , , t , are. Then, for appropriate small values δ k 1 > 0 , the lists Λ k that are not CS1 realizable become

Γ k = { λ k 1 + N C S 1 ( Λ k ) + δ k 1 , λ k 2 , , λ k p k } , k = 2 , r ,

which are now also UR and without loss of generality we assume that they have a diagonalizable positive realizing matrix A k CS λ k 1 + N C S 1 ( Λ k ) + δ k 1 , k = 2 , r . For those lists Λ j , j = r + 1 , , t , that are CS1 realizable, we take Γ j = { λ j 1 + δ j 1 , λ j 2 , , λ j p j } , which are UR with a diagonalizable positive realizing matrix A j CS λ j 1 + δ j 1 , j = r + 1 , , t .

Next, in the first step of the procedure, we apply Theorem 1.4 to the matrices A 1 and A 2 to obtain

(8) M 1 = A 1 A 2 + X C X ˜ ,

which will be diagonalizable positive with spectrum

F 1 = { μ 1 , λ 12 , , λ 1 p 1 } { λ 21 , λ 22 , , λ 2 s 2 } ,

where μ 1 = λ M + ε 1 + ε 2 . To do this, we need to compute a 2-by-2 diagonalizable positive matrix B 1 with spectrum { μ 1 , λ 21 } and diagonal entries β 11 = λ M + ε 1 and β 12 = λ 21 + N C S 1 ( Λ 2 ) + δ 21 . Since

β 11 + β 12 = μ 1 + λ 21 for ε 2 = N C S 1 ( Λ 2 ) + δ 21

and

ρ 1 2 = β 11 β 12 μ 1 λ 21 for ρ 1 = ε 2 ( λ M + ε 1 λ 21 ) ,

then

B 1 = β 11 ρ 1 ρ 1 β 12

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 3 . Again, we need a 2-by-2 diagonalizable positive matrix B 2 with spectrum { μ 2 , λ 31 } and diagonal entries β 21 = λ M + ε 1 + ε 2 and β 22 = λ 31 + N C S 1 ( Λ 3 ) + δ 31 , where μ 2 = λ M + ε 1 + ε 2 + ε 3 . Since

β 21 + β 22 = μ 2 + λ 31 for ε 3 = N C S 1 ( Λ 3 ) + δ 31

and

ρ 2 2 = β 21 β 22 μ 2 λ 31 for ρ 2 = ε 3 ( λ M + ε 1 + ε 2 λ 31 ) ,

then

B 2 = β 21 ρ 2 ρ 2 β 22

is diagonalizable positive, and we obtain a diagonalizable positive matrix M 2 with spectrum

F 2 = { μ 2 , λ 12 , , λ 1 p 1 } { λ 21 , , λ 2 p 2 } { λ 31 , , λ 3 p 3 } ,

where μ 2 = λ M + ε 1 + ε 2 + ε 3 . The procedure follows similar steps for the matrices M k and A k + 2 , k = 3 , , r 2 . Observe that in the ( k 1 ) s t step, we have

ε k = N C S 1 ( Λ k ) + δ k 1 and ρ k 1 = ε k ( λ M + ε 1 + + ε k 1 λ k 1 ) .

For the matrices A j , j = r + 1 , , t , we do the same, except that in this case, we have N C S 1 ( Λ j ) = 0 . Then, the eigenvalues and diagonal entries in the 2-by-2 matrix B j are μ j , λ j + 1 , 1 and β j 1 = λ M + ε 1 + ε 2 + + ε j , β j 2 = λ j + 1 , 1 + δ j + 1 , 1 , respectively, and we have

ε j = δ j 1 and ρ j 1 = ε j ( λ M + ε 1 + + ε j 1 λ k 1 ) .

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

i = 1 t ε i = λ 1 λ M and k = 2 t δ k 1 + ε 1 = λ 1 λ M k = 2 t N C S 1 ( Λ k ) .

Example 3.1

Consider the list

Λ = { 14 , 6 , 5 , 5 , 5 , 1 ± i , 1 ± i } = Λ 1 Λ 2 , with Λ 1 = { 14 , 5 , 5 , 1 ± i } , Λ 2 = { 6 , 5 , 1 ± i } , Γ 1 = { 12 , 5 , 5 , 1 ± i } , Γ 2 = { 7 , 5 , 5 , 1 ± i } .

Observe that Λ is not CS1 realizable, but it is CS2 realizable. In fact, C S 1 ( Λ 1 ) = 2 , T 2 = N C S 1 ( Λ 2 ) = 1 , and λ M = 12 . Then (4) is satisfied. We will show that Λ is UR. Since Λ 1 is CS1 realizable, we obtain for 12 + 1 2 , 5 , 5 , 1 ± i the diagonalizable positive realizing matrix

A 1 = 1 10 1 51 51 11 11 51 1 51 11 11 51 51 1 11 11 1 71 51 1 1 1 51 51 21 1 .

The list 6 + 1 + 1 2 , 5 , 1 ± i has the diagonalizable CS1 realization

A 2 = 1 8 1 41 9 9 41 1 9 9 1 57 1 1 1 41 17 1 .

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 entries 25 2 and 15 2 . It is

B = 25 2 γ γ 15 2 with γ = 39 2 and C = 0 γ γ 0 .

Then,

M = A 1 A 2 + X C X ˜

is diagonalizable positive with spectrum Λ . Hence, Λ is UR.

Remark 3.1

If Λ = { λ 1 , λ 2 , , λ n } is a list of complex numbers of Suleimanova type, that is, λ 1 > 0 , with Re λ i < 0 , Re λ i > Im λ i , i = 1 , 2 , , n , then Λ is not only CS1-realizable but also CS1-UR.

Theorem 3.3

Let Λ = { λ 1 , λ 2 , , λ n } be anE-CS2 realizable list of complex numbers, where λ 1 > λ 2 λ p are real and λ p + 1 , λ p + 2 , , λ n are conjugate complex. If

(9) λ 1 > λ M + k = 2 t N K ( Λ k )

with

λ M = max { λ 1 K ( Λ 1 ) ; max 2 k t { λ k 1 } } ,

then Λ is UR.

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 Λ k , k = 1 , 2 , , t , produces a diagonalizable realizing matrix.□

Example 3.2

Consider the list

Λ = 7 , 2 , 2 , 1 , 1 , 4 , 4 , 1 ± 9 5 i , 1 ± 9 5 i .

It is not CS1 nor CS2, but it is E-CS2 realizable. Let Λ be partitioned as Λ = Λ 1 Λ 2 Λ 3 , where

Λ 1 = { 7 , 1 , 1 , 4 , 4 } and Λ 2 = Λ 3 = 2 , 1 ± 9 5 i .

Since

SR ( Λ 1 ) = 1 , N LL ( Γ 2 ) = N LL ( Γ 3 ) = 9 5 3 3 , λ M = max { 6 , 2 , 2 } = 6 , and λ 1 = 7 > 6 + 2 9 5 3 3 = 18 5 3 ,

where SR and LL are the criteria in [27, Lemma 1] and [24], respectively, then Λ is realizable. For

Γ 1 = { 6 , 1 , 1 , 4 , 4 } ,

we construct, from the necessary and sufficient condition in [27, Lemma 1], the realizing symmetric circulant matrix

A 1 = 0 3 + 5 2 3 5 2 3 5 2 3 + 5 2 3 + 5 2 0 3 + 5 2 3 5 2 3 5 2 3 5 2 3 + 5 2 0 3 + 5 2 3 5 2 3 5 2 3 5 2 3 + 5 2 0 3 + 5 2 3 + 5 2 3 5 2 3 5 2 3 + 5 2 0 .

For Λ 2 = Λ 3 , we have

A 2 = A 3 = 0 2 0 0 0 2 53 25 3 25 0

with spectrum Λ 2 = Λ 3 . Then,

A = A 1 F A 2 F A 3 + eq T , A CS 6 ,

where q T = 0 0 0 0 0 0 1 2 0 0 1 2 0 and

F = 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0

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

4 On the union of two 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 1 Λ 2 is positively DR and therefore UR. Let Λ = { λ 1 , λ 2 , , λ n } and Λ / λ 1 = { λ 2 , , λ n } . In [28] the authors showed that there is a minimal nonnegative number g d ( Λ 1 / α 1 ) , called diagonalizable realizability index, such that Γ = { μ , λ 2 , , λ n } is DR for every μ g d ( Λ 1 / α 1 ) .

Theorem 4.1

Let Λ 1 = { α 1 , α 2 , , α n } and Λ 2 = { β 1 , β 2 , , β m } be UR spectra of complex numbers. Let g d ( Λ 1 / α 1 ) be the diagonalizable realizability index of Λ 1 . If

(10) α 1 = g d ( Λ 1 / α 1 ) + ε and α 1 > β 1 + ε 2 , ε > 0 ,

then Λ 1 Λ 2 is UR.

Proof

Since

{ g d ( Λ 1 / α 1 ) , α 2 , , α n }

is DR, then from [2, Lemma 3.1]  Γ 1 = { g d ( Λ 1 / α 1 ) + ε 2 , α 2 , , α n } is the spectrum of a diagonalizable positive matrix A C S g d ( Λ 1 / α 1 ) + ε 2 . Moreover, there exists a diagonalizable positive matrix B C S β 1 + ε 2 with spectrum

Γ 2 = β 1 + ε 2 , β 2 , , β m .

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 d ( Λ 1 / α 1 ) + ε 2 and β 1 + ε 2 . Since

g d ( Λ 1 / α 1 ) + ε 2 + β 1 + ε 2 = α 1 + β 1

and

g d ( Λ 1 / α 1 ) + ε 2 β 1 + ε 2 ( α 1 β 1 ) = ρ 2

for

ρ = ε 2 α 1 β 1 ε 2 ,

such matrix exists. Then, the matrix

M = A B + X C X ˜

is diagonalizable positive with spectrum Λ 1 Λ 2 . Hence, Λ 1 Λ 2 is UR.□

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.

Corollary 4.1

Let Λ 1 = { α 1 , α 2 , , α n } and Λ 2 = { β 1 , β 2 , , β m } be DR lists of complex numbers. Let g d ( Λ 1 / α 1 ) be the diagonalizable realizability index of Λ 1 . If (10)is satisfied, then Λ 1 Λ 2 is diagonalizably positively realizable, and therefore, Λ 1 Λ 2 is UR.

Proof

The result is immediate from the proof of Theorem 4.1.□

Corollary 4.2

Let Λ 1 = { α 1 , α 2 , , α n } be the spectrum of a diagonalizable nonnegative matrix with a positive row or column, and let Λ 2 = { β 1 , , β m } be DR. If α 1 > β 1 , then Λ 1 Λ 2 is UR.

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 CS α 1 and B CS β 1 . Then, for

F = α 1 β 1 0 0 α 1 β 1 α 1 β 1 0 0 ,

the matrix

M = A 0 F B

is diagonalizable nonnegative with spectrum Λ 1 Λ 2 and with its first column being positive. Hence, Λ 1 Λ 2 is UR.□

Remark 4.1

Theorem 4.1 and Corollaries 4.1 and 4.2 can be useful to decide the universal realizability of some lists. For instance, is the list

Λ = { 7 , 4 , 2 , 2 ± 3 i , 2 ± 3 i } UR?

The lists

Λ 1 = { 7 , 2 , 2 ± 3 i } and Λ 2 = { 4 , 2 ± 3 i }

satisfy Corollary 4.2. Then,

Λ = Λ 1 Λ 2 = { 7 , 4 , 2 , 2 ± 3 i , 2 ± 3 i }

is UR.

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], the authors proved that a realizable list Λ = { λ 1 , λ 2 , , λ n } of real numbers with λ i < 0 , i = 2 , , n , and i = 1 n λ i 0 is UR. It is clear that the list Λ t = { λ 1 + t , λ 2 t , λ 3 , , λ n } , λ i < 0 , i = 2 , , n , 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].

Theorem 4.2

Let Λ = { λ 1 , λ 2 , , λ n } be aCS1 UR list of complex numbers with λ 1 > λ 2 λ p being real and λ p + 1 , λ p + 2 , , λ n being conjugate complex. Let t 0 . Then,

Λ t = { λ 1 + t , λ 2 ± t , λ 3 , , λ n }

is also CS1UR.

Proof

Consider the partition

Λ t = Λ 1 Λ 2 , with Λ 1 = { λ 1 + t , λ 2 ± t , λ 3 , , λ p } , Λ 2 = { λ p + 1 , , λ n } 2 p n .

It is clear that the list

Γ 1 = λ p S k < 0 S k ± t , λ 2 ± t , , λ p

is S1 realizable, and therefore, it is also symmetrically (diagonalizably) realizable. Then, from a Fiedler’s result [26],

Γ 1 , ε = λ p S k < 0 S k ± t + ( ξ λ p ) + ε Only if ξ i > λ p , λ 2 ± t , , λ p

is positively symmetrically realizable. Let B > 0 , B C S β be the realizing matrix for Γ 1 , ε where β = λ p S k < 0 S k ± t + ( ξ λ p ) + ε Only if ξ > λ p . Now, we consider the list Λ 2 of complex numbers. Let

C = C p + 1 , p + 1 C n 1 , n 1 , with C k k = Re λ k Im λ k Im λ k Re λ k , k = p + 2 j + 1 , j = 0 , 1 , , n 2 p 2 .

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

B 0 F C C S β , and A = B 0 F C + eq T

is diagonalizable nonnegative in C S λ 1 + t with spectrum Λ t = { λ 1 + t , λ 2 ± t , λ 3 , , λ n } and with its first column being positive, where q T = [ q 1 , q 2 , , q n ] with

q i = 0 , i = 1 , 2 , , p . q k = min { 0 , Re λ k , Im λ k } , k = p + 1 , , n .

Then, from the extension in [5], Λ t is CS1 UR.□

1. Funding information: This research was supported by Universidad Católica del Norte-VRIDT 036-2020, Nucleo 6 UCN VRIDT-083-2020, Chile.

2. Conflict of interest: The authors declare no conflicts of interest.

3. Data availability statement: All data generated or analyzed during this study are included in this published article (and its supplementary information files).

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