A Note on the Symmetric Nonnegative Inverse Eigenvalue Problem 1

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1 International Mathematical Forum, Vol. 6, 011, no. 50, A Note on the Symmetric Nonnegative Inverse Eigenvalue Problem 1 Ricardo L. Soto and Ana I. Julio Departamento de Matemáticas Universidad Católica del Norte Casilla 180, Antofagasta, Chile Abstract The symmetric nonnegative inverse eigenvalue problem is the problem of characterizing all possible spectra of n n symmetric entrywise nonnegative matrices. The problem remains open for n 5. A number of realizability criteria or sufficient conditions for the problem to have a solution are known. In this paper we show that most of these sufficient conditions can be obtained by the use of a result by Soto, Rojo, Moro, Borobia in [ELA 16 (007) 1-18]. Moreover, by applying this result we may always compute a solution matrix. Mathematics Subject Classification: 15A18 Keywords: nonnegative inverse eigenvalue problem 1 Introduction The symmetric nonnegative inverse eigenvalue problem (hereafter SNIEP) is the problem of finding necessary and sufficient conditions for a list Λ = {λ 1,λ,...,λ n } of real numbers to be the spectrum of an n n symmetric nonnegative matrix. If there exists a symmetric nonnegative matrix A with spectrum Λ, we say that Λ is symmetrically realizable and that A is the realizing matrix. When Λ is a set of real numbers and the realizing matrix A is required to be nonnegative, not necessarily symmetric, the problem is called the real nonnegative inverse eigenvalue problem (RNIEP). This problem remains unsolved. A complete solution is only known for n 4 [9, 11, 5]. Necessary conditions have been found in [9, 7, 1]. However, they are very far from the sufficient conditions, which are known in the literature about the problem. A 1 Supported by Fondecyt , Chile. Corresponding author: rsoto@ucn.cl (R.L.Soto), ajt001@ucn.cl (A.I. Julio).

2 448 R. L. Soto and A. I. Julio number of sufficient conditions for the existence of a solution to the RNIEP are known [4, 1, 13, 15, 6,, 17, 16, 19]. In [10] the authors construct a map of the sufficient conditions for the RNIEP and establish inclusion relations or independency relations between them. Sufficient conditions for the symmetric case (SNIEP) have been obtained in [3, 3, 14, 0, 8, 1]. The RNIEP and the SNIEP are equivalent for n 4 (see [4]), otherwise they are different (see [5]). Our aim in this paper is to show the relevance of the following symmetric realizability criterion given in [1, Theorem 3.1], which we shall call the SRMB criterion. In particular we will show that most of known results, which give realizability criteria for the problem to have a solution, may be obtained by applying the result in [1]. Moreover, the SRMB criterion has a constructive proof, in the sense that it generates an algorithmic procedure to compute a realizing matrix. Theorem 1.1 SRMB criterion [1]Let Λ={λ 1,...,λ n } be a set of real numbers with λ 1 λ... λ n and, for some t n, let ω 1,...,ω t be real numbers satisfying 0 ω k λ 1, k =1,...,t. Suppose there exist i) a partition Λ=Λ 1... Λ t, with Λ k = {λ k1,λ k,...,λ kpk }, λ 11 = λ 1, λ k1 0; λ k1... λ kpk, such that for each k =1,...,t the set Γ k = {ω k,λ k,...,λ kpk } is realizable by a symmetric nonnegative matrix of order p k, and ii) a symmetric nonnegative t t matrix with eigenvalues λ 11,λ 1,...,λ t1 and diagonal entries ω 1,ω,...,ω t. Then Λ is realizable by a symmetric nonnegative matrix of order n. The SRMB criterion was obtained by applying the following result by Soto, Rojo, Moro, Borobia [1, Theorem.6] Theorem 1. [1] Let A be an n n symmetric matrix with eigenvalues λ 1,λ,...,λ n. Let {x 1, x,...,x r } an orthonormal set of eigenvectors of A such that AX = XΩ, where X =[x 1 x x r ] and Ω=diag{λ 1,λ,...,λ r }. Let C be any r r symmetric matrix. Then the symmetric matrix A + XCX T has eigenvalues μ 1,...,μ r,λ r+1,...,λ n, where μ 1,...,μ r are eigenvalues of the matrix Ω+C. Theorem 1. is a symmetric version of a result of Rado, introduced by Perfect in [13], which shows how to modify r eigenvalues of a matrix of order n, via a rank-r perturbation, without changing any of the remaining (n r) eigenvalues.

3 Symmetric nonnegative inverse eigenvalue problem 449 Rado Theorem was applied by Perfect in [13], to derive an efficient realizability criterion for the RNIEP. Surprisingly, this result was somehow ignored in the literature about the problem, until in [19], the authors rescue it and extend it to a new realizability criterion. In [], by using the SRMB criterion, with an special partition of the list Λ, the authors obtain efficient sufficient conditions. Symmetric realizability criteria It is well known that the RNIEP and the SNIEP are equivalent for n 4 [4]), otherwise they are different [5]. In this section, our aim is to show that most of known symmetric realizability criteria can be obtained by applying Theorem 1.1. It was proved in [1] that SRMB criterion contains, strictly, all previous symmetric realizability criteria. The first results about symmetric nonnegative realization are due to Fiedler [3]. ). Several realizability criteria obtained for the RNIEP have later been shown to be realizability criteria for the SNIEP as well. Fiedler [3] and Radwan [14] proved, respectively, that Kellogg [6] and Borobia [] realizability criteria are also symmetric realizability criteria. In [0] it was shown that Soto criterion in [17] is also a symmetric realizability criterion. Fiedler first showed that Suleimanova criterion for the RNIEP, is also a symmetric realizability criterion. Next, by using Theorem 1.1, we give an alternative proof of this fact: Theorem.1 (Symmetric Suleimanova): Let Λ={λ 1,λ,...,λ n } be satisfying λ 1 + λ + + λ n 0 and λ k < 0, k =,...,n. Then Λ is symmetrically realizable. Proof. (SRMB using proof): For n = 1, the result is clear. Suppose n = and Λ = {λ 1,λ } with λ < 0. Consider the partition (as in [, Theorem 4.1]) Λ 1 = {λ 1 }, Λ = {λ } with Γ = {λ 1,λ }. Then, Γ is realizable by the symmetric nonnegative matrix ] A =. [ λ1 +λ λ 1 λ λ 1 λ λ 1 +λ Now we look for a symmetric nonnegative matrix B with eigenvalue λ 1 and diagonal entry λ 1, that is B =[λ 1 ]. Then [ ] [ ] A = A + [0] = A.

4 450 R. L. Soto and A. I. Julio Suppose the result holds for k<n.then there exists a symmetric nonnegative matrix A 1 with spectrum {λ 1,λ,...,λ k }. Let Λ = {λ 1,...,λ k,λ k+1 } and consider the partition Λ 1 = {λ 1,...λ k }, Λ = {λ k+1 }, Λ j = with Γ = {λ λ k,λ k+1 }, Γ j = {0}, j=3,...,k+1. Note that { λ k+1,λ k+1 } is symmetrically realizable and λ 1 + +λ k λ k+1 Let A j,j=,...,k+1, a symmetric nonnegative matrix realizing Γ j. Let M = A A k+1 ; X =[x 1 x k ] with ( ) T x 1 = 0 0, xj = e j+1, j =,...,k. The following sufficient conditions, due to Fiedler [3, Theorem 4.4], guarantee the existence of a symmetric nonnegative matrix with eigenvalues α 1 α α t and diagonal entries ω 1 ω... ω t : Let the vectors α =(α 1,α,...,α t ) and ω =(ω 1,ω,...,ω t ). If i) α majorizes ω and ii) ω k 1 λ k, k =,...,t 1, (1) then there exists a symmetric nonnegative matrix B with eigenvalues α 1,...,α t and diagonal entries ω 1,...,ω t. Then it is clear from (1), that we may construct a symmetric nonnegative matrix B (see [1, Remarks 3.5, 3.7]) with spectrum {λ 1,...,λ k } and diagonal entries λ λ k, 0,...,0. Let C = B diag{λ 1 + +λ k, 0,...,0}. Then A = M +XCX T is symmetric nonnegative with spectrum Λ. Next we show that Perfect criterion in [1] is also a symmetric realizability criterion: Theorem. [1] Let Λ={λ 0,λ 1,λ 11...,λ 1p1,...,λ r,λ r1...,λ rpr,δ}, where λ 0 λ, for λ Λ, λ Λ λ 0, δ 0 λ j 0 and λ ji 0 for j =1,...,r and i =1,...,p j. If p j λ j + δ 0 and λ j + λ ji 0 for j =1,...,r, then Λ is symmetrically realizable.

5 Symmetric nonnegative inverse eigenvalue problem 451 Proof. (SRMB using proof) Let us consider the partition Λ = Λ 0 r Λ i, where Λ 0 = {λ 0,δ}, with Γ 0 = { δ,δ} and Λ i = {λ i,λ i1...,λ ipi },,...,r, with Γ i = { p i λ ij,λ i1...,λ ipi },,...,r. j=1 We may assume, without loss of generality, that λ Λ λ = 0. Clearly, the lists Γ i are symmetrically realizable (they are Suleimanova lists), say by matrices A i,i=0,...r. Then M = A 0 A 1 A r is symmetric nonnegative. Now we need to show that there exists a symmetric nonnegative matrix B with eigenvalues and diagonal entries λ 0,λ 1,λ...,λ r δ, p 1 λ 1j,..., pr λ rj, j=1 respectively. We may re-order these eigenvalues and diagonal entries in such a way that p 1 j=1 j=1 λ 0 λ 1 λ r λ 1j δ pr λ rj, with δ in the corresponding place in between the sequence of p 1 j=1 j=1 λ ij,i= 1,...,r.Since λ Λ λ =0, the sufficient conditions in (1) are satisfied and there exists the required matrix B. Then from Theorem 1., for appropriate matrices X and C, A = M + XCX T is symmetric nonnegative with spectrum Λ. Next, by using Theorem 1., we show that Kellogg criterion is also a symmetric realizability criterion. Besides, different from Kellogg and Fiedler, our proof generate an algorithmic procedure to compute the realizing matrix. Theorem.3 [3] (Symmetric Kellogg) Let Λ={λ 1,λ,...,λ n } be a list of real numbers with λ 1 λ λ n and let p be the greatest index j (1 j n) for which λ j 0. Let the set of indices [ ] n +1 K = {i : λ i 0 and λ i + λ n i+ < 0, i {, 3,... }}.

6 45 R. L. Soto and A. I. Julio If and λ 1 + i K, i<k λ 1 + i K(λ i + λ n i+ )+ (λ i + λ n i+ )+λ n k+ 0 for all k K, () n p+1 j=p+1 λ j 0, provided that n p, (3) then Λ is the spectrum of an n n symmetric nonnegative matrix. Proof. (SRMB using proof). Suppose conditions () and (3) are satisfied and let K = {k1,k,...,kt} be the Kellogg set of indices. Consider the partition (SRMB partition in [1, Theorem 3.1]) Λ = Λ 1 t Λ ki Λ R, where Λ 1 = {λ 1,λ p+1,...,λ n p+1 } Λ ki = {λ ki,λ n ki+ },ki K, i =1,...,t Λ R = Λ Λ 1 t Λ ki, with λ k1 λ k λ kt. It is clear from (3) that Γ 1 = {λ 1 + t (λ ki + λ n ki+ ),λ p+1,...,λ n p+1 } is symmetrically realizable (Observe that Γ 1 is a Suleimanova list). Let A 1 = A T 1 0 realizing Γ 1. We associate, to each list Λ ki, a symmetric realizable list Γ ki = { λ n ki+,λ n ki+ } with realizing symmetric nonnegative matrix A ki,,,...,t. Then M = A 1 A k1 A kt is symmetric nonnegative with spectrum Γ 1 t Γ ki. Observe that elements of the list Λ R, if there exist some-ones, are of the form {λ ki,λ n ki+ } with λ ki +λ n ki+ 0. Then Λ R is symmetrically realizable, say by A R. Now we show that there exists a symmetric nonnegative matrix B, of order (t + 1), with eigenvalues and diagonal entries { λ 1,λ k1,λ k,...,λ kt and λ 1 + t (λ (4) ki + λ n ki+ ), λ n k1+,..., λ n kt+, respectively. To do this, we show that sufficient conditions (1) are satisfied. We know that λ n k1+ λ n k+ λ n kt+ 0.

7 Symmetric nonnegative inverse eigenvalue problem 453 Then we re-order the decreasing sequence of possible diagonal entries by setting ω = λ 1 + t (λ ki + λ n ki+ ) in the right place, say in position r +1. Thus (4) becomes { λ 1,λ k1,λ k,...,λ kt and (5) λ n k1+,...,ω,..., λ n kt+. From (3) we have, respectively for k1,k,...,kt in K, that λ 1 λ n k1+ λ 1 + λ k1 λ n k1+ λ n k+ and λ 1 + λ k1 + λ k λ n k1+ λ n k+ λ n k3+.. λ 1 + s 1 λ ki s λ n ki+, s =1,...,t 1 λ 1 + t (λ ki + λ n ki+ ) t λ n ki+ = λ 1 + λ k1 + + λ kt. Observe that for ω = λ 1 + t (λ ki+λ n ki+ ) in position r+1 in the decreasing sequence of possible diagonal entries, we have and λ 1 + r λ ki λ 1 + r λ ki + t (λ ki + λ n ki+ ) i=r+1 = r λ n ki+ + ω, λ 1 + r+1 λ ki r+1 λ n ki+ + ω. Thus, i) in (1) holds. Finally, since λ n ki+ λ ki for all ki K, and for ω in position r + 1 we have ω λ n k(r+1)+ λ k(r+1), then ii) in (1) holds. Hence B there exists and it can be constructed (see [1, Remarks 3.5, 3.7]). Let M = A k1 A 1 A kt with A 1 in position r +1. Therefore, from Theorem 1., for C = B diagb, the matrix M + XCX T is symmetric nonnegative with spectrum Λ 1 t Λ ki. Hence A = (M + XCX T ) A R is the desired matrix realizing Λ.

8 454 R. L. Soto and A. I. Julio Now we show that Borobia realizability criterion [] is also a symmetric realizability criterion. Different from Borobia and Radwan, our proof generate an algorithmic procedure to construct the realizing matrix. Theorem.4 [14] (Symmetric Borobia): Let Λ={λ 1,λ,...,λ n } be a list of real numbers with λ 1 λ... λ n and let p be the greatest index j (1 j n) for which λ j 0. If there exists a partition J 1 J... J t of J = {λ p+1,λ p+,...,λ n }, for some 1 t n p +1, such that λ 1 λ... λ p λ J 1 λ λ J λ... λ J t λ (6) satisfies the Kellogg conditions () and (3), then Λ is the spectrum of a symmetric nonnegative matrix of order. First we need the following immedite lemma: Lemma.1 [18] Let B k a symmetrix nonnegative matrix realizing { λ n ki+,λ n ki+ } (Suleimanova list), where λ n ki+ = μ μ s 1, μ i < 0,,...,s 1. Then there exists an s s symmetrix nonnegative matrix C k realizing { λ n ki+,μ 1,...,μ s 1 } (Suleimanova list). Proof. Theorem.4 (SRMB using proof). The same proof of Theorem.3 holds here, except for one detail: the list in (6) has less elements than the original list Λ and we need to obtain an n n symmetric nonnegative matrix realizing Λ. Then, before to apply Theorem 1., we apply, if it is necessary, to each matrix A ki realizing Γ ki = { λ n ki+,λ n ki+ },ki K (and possibly to the matrix A 1 realizing Γ 1 ), the Lemma.1 to obtain a symmetric nonnegative matrix Âki with spectrum { λ n ki+,μ 1,...,μ s 1 }, where λ n ki+ = μ μ s 1,μ i < 0,,...,s 1. Next, we consider the following result of Laffey-Šmigoc [8], and give an alternative proof of it. Theorem.5 [8] Let A be an n n irreducible symmetric nonnegative matrix with spectrum {λ 1,...,λ n } and a diagonal c. Let B be an m m symmetric nonnegative matrix with spectrum {μ 1,...,μ m }. i) If μ 1 c, then there exists a symmetric nonnegative matrix C, of order n + m 1, with spectrum {λ 1,...λ n,μ,...,μ m }. ii) If c μ 1, then there exists a symmetric nonnegative matrix C, of order n + m 1, with spectrum {λ 1 + μ 1 c, λ,...λ n,μ,...,μ m }.

9 Symmetric nonnegative inverse eigenvalue problem 455 Proof. (SRMB using proof): i) Let Λ = {λ 1,...λ n,μ,...,μ m } and consider the partition (as in [, Theorem 4.1]) Λ = Λ 1 Λ Λ n+1, where Λ 1 = {λ 1,...,λ n } Λ = {μ,...,μ m } Λ k =, k =3,...,n+1. We know from the hypothesis that Λ 1 is symmetrically realizable by A. Let c, c,...c n be the diagonal entries of A and consider the associated symmetrically realizable lists Γ = {c, μ,...,μ m }; Γ k = {c k 1 }, k =3,...,n+1. Since c μ 1 and {μ 1,...,μ m } is the spectrum of B (symmetric nonnegative), and c k 1 0, then all list Γ k are symmetrically realizable. Let A be symmetric nonnegative realizing Γ and let [c k 1 ] be the 1 1 matrix realizing Γ k,k= 3,...,n+1. Then M = A [c ] [c n ] is symmetric nonnegative of order n + m 1 with spectrum {c, μ,...,μ m,c,...,c n }. Now, since A is symmetric nonnegative with spectrum Λ 1 and diagonal entries c, c,...c n, then from Theorem 1. with C = A diag{c, c,...c n } and X as in Theorem 1., the matrix M +XCX T is symmetric nonnegative with spectrum Λ. ii) Now, let c μ 1 and let Λ = {λ 1 + μ 1 c, λ,...λ n,μ,...,μ m }. Consider the partition (as in [, Theorem 4.1]) Λ = Λ 1 Λ Λ n+1, where Λ 1 = {λ 1 + μ 1 c, λ,...,λ n } Λ = {μ,...,μ m } Λ k =, k =3,...,n+1. It is clear that Λ 1 is symmetrically realizable since μ 1 c 0. Consider the associated symmetrically realizable lists Γ = {μ 1,μ,...,μ m }; Γ k = {c k 1 }, k =3,...,n+1.

10 456 R. L. Soto and A. I. Julio We know that the symmetric nonnegative matrix B realizes Γ and [c k 1 ] realizes Γ k,k=3,...,n+1. Let M = B [c ] [c n ]. Now we apply Lemma 6 in [8] to the matrix A (with t = μ 1 c) to obtain a symmetric nonnegative matrix A 0 with spectrum Λ 1 and diagonal entries μ 1,c,...,c n. Then from Theorem 1., with C = A 0 diag{μ 1,c,...c n }, the matrix M + XCX T is symmetric nonnegative with spectrum Λ. In [14], the author presents the following unpublished result due to Loewy, which we prove here by using SRMB result: n Theorem.6 Let n 4, λ 1 λ 0 λ 3 λ n, λ i 0. Suppose K 1,K is a partition of {3, 4,...,n} such that λ 1 λ i i K 1 λ i. Then there exists a symmetric nonnegative matrix with spectrum i K Λ={λ 1,λ,...,λ n }. Proof. Let n λ i = s 0. Consider the partition Λ = Λ 0 Λ 1 Λ, where Λ 0 = {λ 1,λ }, Λ 1 = {λ i : i K 1 } = {λ 11,λ 1,...,λ 1r }, Λ = {λ i : i K } = {λ 1,λ,...,λ t }, r + t = n. with symmetrically realizable associated lists (Suleimanova lists) Γ 1 = {ω 1,λ 11,λ 1,...,λ 1r }; Γ = {ω,λ 1,λ,...,λ t } where ω 1 = λ i + α, ω = λ i + β with α + β = s, 0 α, β s, i K 1 i K in such a way that 0 ω 1,ω λ 1. Let A 1 and A symmetric nonnegative matrices realizing Γ 1 and Γ, respectively. Since ω 1 + ω = λ 1 + λ, then there exists a symmetric nonnegative matrix B with eigenvalues λ 1,λ and diagonal entries ω 1,ω. Thus, from Theorem 1., with C = B diagb, [ ] A1 0 A = + XCX T 0 A is symmetric nonnegative with spectrum Λ. Next we give an alternative proof of the following result of Fiedler:

11 Symmetric nonnegative inverse eigenvalue problem 457 Lemma. (Fiedler, [3]). Let A be a symmetric m m matrix with spectrum Λ 1 = {α 1,...,α m }. Let u, u =1, be a unit eigenvector of A corresponding to α 1. Let B be a symmetric n n matrix with spectrum Λ = {β 1,...,β n }. Let v, v =1, be a unit eigenvector of A corresponding to β 1. Then for any ρ, the matrix ( ) A ρuv T C = ρvu T B has spectrum Λ={γ 1,γ,α,...,α m,β,...,β n }, where γ 1,γ are eigenvalues of the matrix ( ) α1 ρ Ĉ =. ρ β 1 ( ) A 0 Proof. (SRMB using proof). The matrix M = is symmetric of 0 B order (m + n) with eigenvalues α 1,...,α m,β 1,...,β n. Let x 1 = (u 1,...,u m, 0,...,0) T, x = (0,...,0,v 1,...,v n ) T (m + n) dimensional vectors, where the u i s and the v i s are the entries of the vectors u and v, respectively. Let Ω = diag{α 1,β 1 }, and X =[x 1 x ], where the columns x 1, x form an orthonormal set. Then AX = XΩ. Define the symmetric matrix C = ( 0 ρ ρ 0 ). Then ( 0 ρuv XCX T T = ρvu T 0 ) and ( A ρuv M + XCX T T = ρvu T B ). From Theorem 1., M + XCX T is symmetric with spectrum Λ={γ 1,γ,α,...,α m,β,...,β n }, where γ 1 and γ are eigenvalues of the matrix ( ) α1 ρ Ω+C = ρ β 1

12 458 R. L. Soto and A. I. Julio for any ρ. Observe that if the matrices A and B are symmetric nonnegative and ρ>0, then M + XCX T is also symmetric nonnegative. Theorem 1. also allow us to generalize Lemma.. In fact, if we have symmetric matrices A 1,A,...,A p, with corresponding spectra Λ i = {α (i) 1,α (i),...,α (i) ni },,,...,p and uni-, then we tary eigenvectors u (i) associated, respectively, to the eigenvalues α (i) 1 may obtain a symmetric n n matrix with spectrum {γ 1,...,γ p,α (1) A =(A 1 A... A p )+XCX T,,...,α(1) n1,...,α(p),...,α(p) np }, where γ 1,...,γ p are eigenvalues of the matrix Ω + C, with Ω = diag{α (1) 1,α () 1,...,α (p) 1 }. References [1] G. Bharali, O. Holtz, Functions preserving nonnegativity of matrices, SIAM J. Matrix Appl. 30 (008) [] A. Borobia, On the Nonnegative Eigenvalue Problem, Linear Algebra Appl. 3-4 (1995) [3] M. Fiedler, Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl. 9 (1974) [4] W. Guo, An inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl. 49 (1996) [5] C. R. Johnson, T. J. Laffey, R. Loewy, The real and the symmetric nonnegative inverse eigenvalue problems are different, Proc. AMS 14 (1996) [6] R. Kellogg, Matrices similar to a positive or essentially positive matrix, Linear Algebra Appl. 4 (1971) [7] T. J. Laffey, E. Meehan, A refinament of an inequality of Johnson, Loewy and London on nonnegative matrices and some applications, Electron. J. of Linear Algebra 3 (1998) [8] T. J. Laffey, H. Šmigoc, Construction of nonnegative symmetric matrices with given spectrum, Linear Algebra Appl. 41 (007) [9] R. Loewy, D. London, A note on an inverse problem for nonnegative matrices, Linear and Multilinear Algebra 6 (1978)

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14 460 R. L. Soto and A. I. Julio [5] J. Torre-Mayo, M.R. Abril-Raymundo, E. Alarcia-Estévez, C. Marijuán, M. Pisonero, The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial, EBL digraphs, Linear Algebra Appl. 46 (007) Received: April, 011