REALIZABILITY BY SYMMETRIC NONNEGATIVE MATRICES

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1 Proyecciones Vol. 4, N o 1, pp , May 005. Universidad Católica del Norte Antofagasta - Chile REALIZABILITY BY SYMMETRIC NONNEGATIVE MATRICES RICARDO L. SOTO Universidad Católica del Norte, Chile Received : March 005, Accepted : May 005. Abstract Let Λ = {λ 1,λ,...,λ n } be a set of complex numbers. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions in order that Λ may be the spectrum of an entrywise nonnegative n n matrix. If there exists a nonnegative matrix A with spectrum Λ we say that Λ is realized by A. If the matrix A must be symmetric we have the symmetric nonnegative inverse eigenvalue problem (SNIEP). This paper presents a simple realizability criterion by symmetric nonnegative matrices. The proof is constructive in the sense that one can explicitly construct symmetric nonnegative matrices realizing Λ. AMS classification: 15A18, 15A51. Key words: symmetric nonnegative inverse eigenvalue problem. Supported by Fondecyt , Chile.

2 66 Ricardo L. Soto 1. Introduction The nonnegative inverse eigenvalue problem (hereafter NIEP) is the problem of characterizing all possible spectra of entrywise nonnegative matrices (References [1-17]). This problem remains unsolved. In the general case, when the possible spectrum Λ is a set of complex numbers, the problem has only been solved for n = 3 by Loewy and London [8]. The cases n =4and n = 5 have been solved for matrices of trace zero by Reams [11] and Laffey and Meehan [7], respectively. When Λ is a set of real numbers (RNIEP), sufficient conditions have been obtained in [16], [9], [1], [6], [1], [13]. If Λ has to be the spectrum of a symmetric nonnegative matrix, we have the symmetric nonnegative inverse eigenvalue problem (SNIEP), which is the subject of this paper. AsetΛ of real numbers is said to be realizable if Λ is the spectrum of an entrywise nonnegative matrix. A set K of conditions is said to be a realizability criterion if any set of real numbers Λ = {λ 1,λ,..., λ n } satisfying the conditions K is realizable. In ([13], Theorem 11) the author gives a simple realizability criterion for the existence of an n n nonnegative matrix with real prescribed spectrum. The goal of this work is to show that this criterion is also a realizability criterion for the symmetric nonnegative inverse eigenvalue problem. Unlike several of the previous conditions which are sufficient for realizability of spectra, the proof of Theorem 11 in [13] is constructive in the sense that one can explicitly construct nonnegative matrices realizing the prescribed real spectrum. This is done by employing an extremely useful result, due to Brauer [3], which shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. In [4] Fiedler obtain some necessary and some sufficient conditions for asetofn real numbers Λ = {λ 1,λ,...,λ n } to be the spectrum of an n n symmetric nonnegative. There, Fiedler also shows that Kellogg s realizability criterion [6] is sufficient for the existence of a symmetric nonnegative matrix with prescribed spectrum. In [10], Radwan shows that Borobia s realizability criterion [1] is also sufficient for the existence of a symmetric nonnegative matrix with prescribed spectrum. Soules [15] gives a realizability criterion for the existence of a symmetric doubly stochastic matrix and

3 Realizability by symmetric nonnegative matrices 67 also shows how to construct a realizing matrix. Radwan, in [10], point out that the realizability criteria of Kellogg and Soules are not comparable. In [5], the authors show that the real nonnegative inverse eigenvalue problem and the symmetric nonnegative inverse eigenvalue problem are different, while Wuwen, in [17], shows that both problems are equivalent for n 4. This paper is organized as follows: In section, we introduce the notation and previous results, which will be necessary in order to prove Theorem 1 in section 3. In section 3 we prove that Soto s realizability criterion ([13], Theorem 11) established here as Theorem 1, is sufficient for the existence of an n n symmetric nonnegative matrix with prescribed spectrum. In section 4 we consider the problem of constructing symmetric nonnegative matrices realizing spectra, which satisfy Theorem 1. Some examples are given in section 5.. Preliminaries and notation Following the notation in [], the set A {Λ = {λ 1,λ,...,λ n } R : λ 1 λ i,i=,...,n} includes all possible real spectra of nonnegative matrices. We denote AR={Λ A : Λ is realizable}. We denote by N n the set of all Λ = {λ 1,λ,...,λ n } AR, where λ 1 λ... λ n. Similarly, we denote by S n ( S c n )thesetofallλ N n for which there exists an n n symmetric nonnegative (positive) matrix with spectrum Λ. We shall only consider real sets Λ = {λ 1,λ,...,λ n } satisfying λ 1 λ... λ p 0 >λ p+1... λ n, since if λ n 0, then A = diag{λ 1,λ,...,λ n } is a symmetric nonnegative matrix. The following result, due to Fiedler, shows that if A and B are symmetric matrices of order n and m, respectively, then we may construct a new symmetric matrix of order n + m as follows:

4 68 Ricardo L. Soto Lemma 1. (Fiedler [4]) Let A be a symmetric n n matrix with eigenvalues α 1,α,...,α n. Let u, kuk =1, be a unit eigenvector of A corresponding to α 1. Let B be a symmetric m m matrix with eigenvalues β 1,β,...,β m. Let v, kvk =1, be a unit eigenvector of B corresponding to β 1. Then for any ρ the matrix Ã! A ρuv T C = ρvu T B has eigenvalues α,...,α n,β,...,β m,γ 1,γ, where γ 1 and γ are eigenvalues of the matrix Ã! α1 ρ bc =. ρ β 1 The next relevant result, due also to Fiedler [4], is necessary for the proof of the main result in section 3. Here we present the Wuwen version of it [17]: Lemma. (Fiedler [4]) If {α 1,α,...,α n } S n, {β 1,β,...,β m } S m and ε max{0,β 1 α 1 }, then {α 1 + ε, β 1 ε, α,...,α n,β,...,β m } S n+m. We shall also need the following lemma: Lemma 3. (Fiedler [4]) If Λ = {λ 1,λ,...,λ n } S n and if ε>0 then Λ ε = {λ 1 + ε, λ,...,λ n } b S n. In ([13], Theorem 11) we give the following simple realizability criterion, which also shows how to construct a realizing matrix. Theorem 1. (Soto [13]) Let Λ = {λ 1,λ,...,λ n } be a set of real numbers, such that λ 1 λ... λ p 0 >λ p+1... λ n. If (.1) λ 1 λ n X S k S k <0 where S k = λ k + λ n k+1,k=, 3,..., n and S n+1 =min{λn+1, 0} for n odd, then Λ is realized by a nonnegative matrix A (with constant row sums).

5 Realizability by symmetric nonnegative matrices 69 Observe that if Λ = {λ 1,λ,...,λ n } satisfies the sufficient condition (.1), then Λ 0 = { λ n X S k,λ,...,λ n } is a realizable set. S k <0 3. Realizability by a symmetric nonnegative matrix In this section we show that the realizability criterion given by Theorem 1 is sufficient for the existence of a symmetric nonnegative matrix with prescribed spectrum Λ. Theorem 1. Let Λ = {λ 1 ; λ,...,λ n } be a set of real numbers such that λ 1 λ... λ p 0 >λ p+1... λ n. If Λ satisfies the realizability criterion given by Theorem 1, then Λ is realized by an n n symmetric nonnegative matrix. Proof. is, Suppose that Λ satisfies the condition (.1) of Theorem 1. That λ 1 λ n P S i <0 S i, where S k = λ k + λ n k+1,k=, 3,..., n and S n+1 =min{λn+1, 0} for n odd. It suffices to prove the statement for λ 1 = λ n P S k <0 S k. In fact, if λ 1 > λ n P S k <0 S k then we take Λ e = {µ 1,λ,...,λ n } with µ 1 = λ n P S k <0 S k. Thus, if Λ e S n then we apply Lemma 3 with ε = λ 1 µ 1 > 0 to show that Λ S n (actually Λ S c n ). Let Λ n+1 Consider the partition Λ k = {λ k,λ n k+1 }; k =1,,..., = {λ n+1 } for n odd. n and

6 70 Ricardo L. Soto Λ = [ n ] k=1 Λ k with Λ = [ n ] k=1 Λ k Λ n+1 for n odd. Observe that some subsets Λ k can be realizable thenselves, in particular by the symmetric nonnegative matrix B k = 1 Ã! λk + λ n k+1 λ k λ (3.1) n k+1. λ k λ n k+1 λ k + λ n k+1 Without loss of generality we may reorder the subsets Λ k, in such a way that Λ, Λ 3,...,Λ t, t n, are nonrealizable (Sk < 0), while Λ t+1,...,λ [ n ] are realizable (S k 0). Consider, if there is someone, the realizable sets Λ k :IfB k in (3.1) realizes Λ k, then the direct sum B = B k, ³ = λ n+1 0forn odd, is a sym- k = t +1,..., n, with B n+1 metric nonnegative matrix realizing [ n ] if λ n+1 k=t+1 Λ k ( [ n ] for n k=t+1 Λ k Λ n+1 odd). Now we consider, if there is someone, the nonrealizable sets Λ k,k=, 3,...,t together with the realizable set Λ 1 = {λ 1,λ n } and we renumber the t elements in Λ k as λ 1 λ... λ t λ t+1... λ t 1 λ t. For each one of these sets Λ k,k=1,,...,t, we define the associated set (3.) Γ k = { λ t k+1,λ t k+1 }, which is realizable by the symmetric nonnegative matrix (3.3) A k = Ã 0 λ t k+1 λ t k+1 0! with Γ t+1 = {0} if λ t+1 < 0forn odd, which is realized by the symmetric nonnegative matrix A t+1 =(0). Now, we procede as follows: First, we merge the sets Γ 1 = { λ t,λ t } S and Γ = { λ t 1,λ t 1 } S

7 Realizability by symmetric nonnegative matrices 71 to obtain, from Lemma, a new set S 4. In fact, we take ε = S = (λ + λ t 1 ) > 0. Then and λ t + ε = λ t S = λ t (λ + λ t 1 ) λ t 1 ε = λ t 1 + S = λ t 1 +(λ + λ t 1 )=λ = { λ t S,λ,λ t 1,λ t } S 4. Next we merge with Γ 3 = { λ t,λ t }. Let ε 3 = S 3 = (λ 3 + λ t ) > 0. Then λ t S + ε 3 = λ t S S 3 λ t ε 3 = λ t + S 3 = λ t +(λ 3 + λ t )=λ 3 and from Lemma 3 = { λ t S S 3,λ 3,,..., } S 6. Observethatineachstepwerecoverthefirst element λ k Λ k from λ t k+1 ε k = λ k. In the j th step of the procedure ( j ), we merge the sets j = { λ t S S 3 S j,λ j,,..., } and Γ j+1 = { λ t j,λ t j }. Then for ε j+1 = S j+1 = (λ j+1 + λ t j ) > 0wehave jx j+1 X λ t S k + ε j+1 = λ t S k k= k= λ t j ε j+1 = λ j+1 and from Lemma j+1 = { λ t P j+1 k= S k,λ j+1,,..., } S j+. In the last step ((t 1) step) we merge the sets t 1 X t 1 = { λ t S k,λ t 1,,..., } S t k= Γ t = { λ t+1,λ t+1 }. and

8 7 Ricardo L. Soto Let ε t = S t = (λ t + λ t+1 ). Then from Lemma we obtain tx t = { λ t S k,λ t,,..., } k= = {λ 1,λ,...,λ t,λ t+1,...,λ t 1,λ t } S t. Now, if n is odd with λ t+1 to obtain < 0thenwealsomerge t with Γ t+1 = {0} 0 t = { λ t tx k= = {λ 1,...,λ t,λt+1 S k S t+1,λt+1,λ t,,..., },λ t+1,...,λ t } S t+1. Thus, if A is a symmetrix nonnegative matrix realizing t = t k=1 Λ k ( 0 t = t k=1 Λ k Λ n+1 ), then A B realizes Λ = {λ 1 ; λ,...,λ n }. That is Λ S n. 4. Constructing the realizing matrix Let Λ = {λ 1,λ,...,λ n } be as in Theorem 1 with λ 1 = λ n P S k <0 S k. Consider the partition Λ = [ n ] k=1 Λ k with Λ = [ n ] k=1 Λ k Λ n+1 where Λ k = {λ k,λ n k+1 } ; k =1,,..., n and Λ n+1 For k =, 3,..., n let A = {Λ k : S k = λ k + λ n k+1 < 0} B = {Λ k : S k = λ k + λ n k+1 0}. for n odd, n = λ n+1 o for n odd. Note that A or B can be empty, n 3, and Λ n+1 can be in A.or B. Each set Λ k B is realizable in particular by the symmetric nonnegative ³ matrix B k in (3.1). Then the direct sum B = B k, with B n+1 = λ n+1 if λ n+1 0forn odd, is a symmetric nonnegative matrix realizing Λ k with Λ k B. Now we consider the nonrealizable sets Λ k A, which can be numbered as Λ, Λ 3,...,Λ t,t n with Λ t+1 = {λ t+1 } if λ t+1 < 0 for n odd. For each Λ k A we define the associated set Γ k,k=, 3,...,t,

9 Realizability by symmetric nonnegative matrices 73 as in (3.) and Γ 1 = { λ t,λ t }, which are realizable in particular by the symmetric nonnegative matrix A k,k=1,,...,t, as in (3.3). As in the proof of Theorem 1 we merge the sets Γ 1 and Γ to obtain = { λ t S,λ,λ t 1,λ t } S 4. Then a symmetric nonnegative matrix which realizes is Ã! A1 ρ M 4 = v u T ρ u v T A where v T = ut =( 1, 1 )andρ = p (λ + λ t )(λ + λ t 1 ). Next we merge with Γ 3 to obtain 3 = { λ t S S 3,λ,λ 3,λ t,λ t 1,λ t } S 6, which, according to Lemma 1, is realized by the symmetric nonnegative matrix Ã! M4 ρ M 6 = 3 v 3 u T 3 ρ 3 u 3 v3 T, A 3 where M 4 v 3 =( λ t S )v 3, kv 3 k =1 and A 3 u 3 =( λ t )u 3, ku 3 k =1 and ρ 3 must be such that Ã! λt S C 3 = ρ 3 ρ 3 λ t has eigenvalues λ t S S 3 and λ 3. The process shows that, in the (k 1) step, we may compute the matrix à Mk ρ M k = k v k u T k ρ k u k vk T A k!, k =, 3,...,t, where M k is the symmetric nonnegative matrix with spectrum k 1,v k and u k are unit eigenvectors of M k and A k, respectively, corresponding to the eigenvalues λ t P k 1 j= S j and λ t k+1, respectively,.and ρ k must be such that the matrix à λt P k 1 C k = j= S! j ρ k ρ k λ t k+1 has eigenvalues λ t P k j= S j and λ k.

10 74 Ricardo L. Soto Now we compute symmetric nonnegative matrices with spectrum Λ for n =4andn =5. Let Λ = {λ 1,λ,λ 3,λ 4 } satisfying the realizability criterion of Theorem 1. We have two cases: i) λ 1 λ 4 with λ + λ 3 0. Then A = 1 λ 1 + λ 4 λ 1 λ λ 1 λ 4 λ 1 + λ λ + λ 3 λ λ λ λ 3 λ + λ 3 ii) λ 1 λ 4 (λ + λ 3 ). Then 0 λ 4 ρ ρ A = 1 λ 4 0 ρ ρ ρ ρ 0 λ 3, ρ ρ λ 3 0 where ρ = λ 3 λ 4 λ 1 λ. Let Λ = {λ 1,λ,λ 3,λ 4,λ 5 } satisfying the realizability criterion of Theorem 1. We have four cases: i) λ 1 λ 5 with λ + λ 4 0andλ 3 0. Then A = 1. λ 1 + λ 5 λ 1 λ λ 1 λ 5 λ 1 + λ λ + λ 4 λ λ λ λ 4 λ + λ λ 3 ii) λ 1 λ 5 (λ + λ 4 )withλ 3 0. Then 0 λ 5 ρ ρ 0 A = 1 λ 5 0 ρ ρ 0 ρ ρ 0 λ 4 0, ρ ρ λ λ 3 where ρ = λ 4 λ 5 λ 1 λ..

11 Realizability by symmetric nonnegative matrices 75 iii) λ 1 λ 5 λ 3 with λ + λ 4 0. Then A = 1 0 λ 5 ρ 0 0 λ 5 0 ρ 0 0 ρ ρ 0 0 0, λ + λ 4 λ λ λ λ 4 λ + λ 4 where ρ = λ 1 λ 3. iv) λ 1 λ 5 (λ + λ 4 ) λ 3. Then A = 1 0 λ 5 ρ ρ λ 5 0 ρ ρ ηv ρ ρ 0 λ 4 ρ ρ λ 4 0 ηv T 0, where v T =(v 1,v,v 3, v 4 )withv 1 = v = p p +q, v 3 = v 4 = q p +q, p = µ 1 +λ 4 +ρ, q = µ 1 +λ 5 +ρ, µ 1 = λ 5 (λ +λ 4 ), ρ = λ 4 λ 5 λ 1 λ and η = λ 1 λ Examples Example 1. Let Λ = {9, 5, 3, 3, 5, 5, 5, 5}. We have the partition Λ = Λ 1 Λ Λ 3 Λ 4, where Λ 1 = {9, 5}, Λ = {3, 5}, Λ 3 = {3, 5} and Λ 4 = {5, 5}. We define the associated sets Γ 1 = {5, 5}, Γ = {5, 5} and Γ 3 = {5, 5}. Then we merge Γ 1 with Γ to obtain A 4 = havig spectrum = {7, 3, 5, 5}. Next we merge with Γ 3 and obtain A 6 =

12 76 Ricardo L. Soto with spectrum 3 = {9, 3, 3, 5, 5, 5}. Finally we have A 8 = with spectrum Λ S 8. Example. Let Λ = {7, 5, 1, 3, 4, 6}. Observe that Λ does not satisfy Theorem 1. However we still may obtain a symmetrix nonnegative matrix realizing Λ : Consider the partition Λ = Λ 1 Λ, where Λ 1 = {7, 6} and Λ = {5, 1, 3, 4}. Define Γ 1 = {6, 6} and Γ = {6, 1, 3, 4}. Then A = realizes Γ while Ã! 0 6 A 1 realizes Γ By applying Lemma to Γ and Γ 1 we obtain Λ andfromlemma1we may compute the realizing matrix A = q q q q q q 3 3 q 0 q

13 Realizability by symmetric nonnegative matrices 77 References [1] A. Borobia, On the Nonnegative Eigenvalue Problem, Linear Algebra Appl. 3-4, pp , (1995). [] A. Borobia, J. Moro, R. Soto, Negativity compensation in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 393, pp , (004). [3] A. Brauer, Limits for the characteristic roots of a matrix. IV: Aplications to stochastic matrices, Duke Math. J., 19, pp , (195). [4] M. Fiedler, Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl. 9, pp , (1974). [5]C.R.Johnson,T.J.Laffey, R. Loewy, The real and the symmetric nonnegative inverse eigenvalue problems are different, Proc. AMS 14, pp , (1996). [6] R. Kellogg, Matrices similar to a positive or essentially positive matrix, Linear Algebra Appl. 4, pp , (1971). [7] T. J. Laffey, E. Meehan, A characterization of trace zero nonnegative 5x5 matrices, Linear Algebra Appl. pp , pp , (1999). [8] R. Loewy, D. London, A note on an inverse problem for nonnegative matrices, Linear and Multilinear Algebra 6, pp , (1978). [9] H. Perfect, Methods of constructing certain stochastic matrices, Duke Math. J. 0, pp , (1953). [10] N. Radwan, An inverse eigenvalue problem for symmetric and normal matrices, Linear Algebra Appl. 48, pp , (1996). [11] R. Reams, An inequality for nonnegative matrices and the inverse eigenvalue problem, Linear and Multilinear Algebra 41, pp , (1996). [1] F. Salzmann, A note on eigenvalues of nonnegative matrices, Linear Algebra Appl. 5., pp , (197).

14 78 Ricardo L. Soto [13] R. Soto, Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra Appl. 369, pp , (003). [14] R. Soto, A. Borobia, J. Moro, On the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem, Linear Algebra Appl. 396, pp. 3-41, (005). [15] G. Soules, Constructing symmetric nonnegative matrices, Linear and Multilinear Algebra 13, pp , (1983). [16] H. R. Suleimanova, Stochastic matrices with real characteristic values, Dokl. Akad. Nauk SSSR 66, pp , (1949). [17] G. Wuwen, An inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl. 49, pp , (1996). Ricardo L. Soto Departamento de Matemáticas Universidad Católica del Norte Casilla 180 Antofagasta Chile rsoto@ucn.cl