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The Inverse Eigenvalue Problem for Symmetric Doubly Stochastic Matrices

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The Inverse Eigenvalue Problem for Symmetric Doubly Stochastic Matrices Linear Algebra and its Applications 379 (2004) 77–83 www.elsevier.com/locate/laa The inverse eigenvalue problem for symmetric doubly stochastic matrices Suk-Geun Hwang a,∗,1, Sung-Soo Pyo b,2 aDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea bSchool of Electrical Engineering and Computer Science, Kyungpook National University, Taegu 702-701, Republic of Korea Received 22 July 2002; accepted 24 December 2002 Submitted by F. Uhlig Abstract s × For a positive integer n and for a real number s, let n denote the set of all n n real matrices whose rows and columns have sum s. In this note, by an explicit constructive method, we prove the following. T λ1 (i) Given any real n-tuple = (λ1,λ2,...,λn) , there exists a symmetric matrix in n whose spectrum is . T (ii) For a real n-tuple = (1,λ2,...,λn) with 1 λ2 ··· λn, if 1 λ λ λn + 2 + 3 +···+ 0, n n(n − 1) n(n − 2) 2 · 1 then there exists a symmetric doubly stochastic matrix whose spectrum is . The second assertion enables us to show that for any λ2,...,λn ∈[−1/(n − 1), 1], there T is a symmetric doubly stochastic matrix whose spectrum is (1,λ2,...,λn) and also that any number β ∈ (−1, 1] is an eigenvalue of a symmetric positive doubly stochastic matrix of any order. © 2003 Elsevier Inc. All rights reserved. ∗ Corresponding author. Tel.: +82-53-950-5887; fax: +82-53-950-6811. E-mail addresses: [email protected] (S.-G. Hwang), [email protected] (S.-S. Pyo). 1 Supported by Com2MaC-KOSEF. 2 This work was done while the second author had a BK21 Post-Doc. position at Kyungpook National University. 0024-3795/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0024-3795(03)00366-5 78 S.-G. Hwang, S.-S. Pyo / Linear Algebra and its Applications 379 (2004) 77–83 AMS classification: 15A51; 15A18 Keywords: Doubly stochastic matrix; Inverse eigenvalue problem; Spectrum 1. Introduction A real matrix A is called nonnegative (resp. positive), written A O (resp. A> O), if all of its entries are nonnegative (resp. positive). A square nonnegative matrix is called doubly stochastic if all of its rows and columns have sum 1. The set of all n × n doubly stochastic matrices is denoted by n. For a square matrix A, let σ (A) denote the spectrum of A. Given an n-tuple T = (λ1,λ2,...,λn) of numbers, real or complex, deciding the existence of a ma- trix A with some specific properties such that σ (A) = has long time been one of the problems of main interest in the theory of matrices. T Given = (λ1,λ2,...,λn) , in order that = σ (A) for some positive matrix A, it is necessary that λ1 + λ2 +···+λn > 0. Sufficient conditions for the exis- tence of a positive matrix A with σ (A) = have been investigated by several au- thors such as Borobia [1], Fiedler [2], Kellog [4], and Salzmann [5]. In this paper we deal with the existence of certain real symmetric matrices and that of symmetric doubly stochastic matrices with prescribed spectrum under certain conditions. Let A ∈ n. Then the well known Gershgorin’s theorem (see [3], for instance) implies that σ (A) is contained in the closed unit disc centered at the origin in the complex plane. Thus if, in addition, A is symmetric, then σ (A) lies in the real closed interval [−1, 1]. T Given a real n-tuple = (λ1,λ2,...,λn) with λ1 λ2 ··· λn, in order that there exists A ∈ n with σ (A) = , it is necessary that λ1 = 1,λn −1,λ1 + λ2 +···+λn 0. (1) Thus, for instance, there exists no 3 × 3 doubly stochastic matrix with spectrum (1, −0.5, −0.6)T or (1, 1, −1.1)T. T Certainly the condition (1) is not sufficient for = (λ1,λ2,...,λn) with λ1 λ2 ··· λn to be the spectrum of a doubly stochastic matrix. T In this paper, we find a sufficient condition on = (λ1,λ2,...,λn) with λ1 λ2 ··· λn satisfying (1) for the existence of a symmetric doubly stochastic matrix having as its spectrum by a constructive method. For an n-tuple x = (x1, T ←− x2,...,xn) , let x and x be defined by T x = (x1 − x2,x2 − x3,...,xn−1 − xn,xn) , ←− T x = (xn,xn−1,...,x2,x1) . S.-G. Hwang, S.-S. Pyo / Linear Algebra and its Applications 379 (2004) 77–83 79 ←− Note that x is the difference sequence of the finite sequence 0, xn,xn−1, ..., x2,x1. Let 1 1 1 T h = 1, , ,..., . n 2 3 n Then 1 1 1 1 T h = , ,..., , . n 1 · 2 2 · 3 (n − 1) · n n We see that the ith component of hn is the length of the ith subinterval of the [ ] − 1 1 1 division of the unit interval 0, 1 by inserting the n 1 points 2 , 3 ,..., n . s × For a real number s, let n denote the set of all n n real matrices all of whose 1 rows and columns have sum s. The set n consists of all nonnegative matrices in n. T In what follows we show that for any real n-tuple = (λ1,λ2,...,λn) , there λ1 exists a symmetric matrix in n whose spectrum is , and also that if a real n-tuple T = (1,λ2,...,λn) with 1 λ2 ··· λn satisfies (1) and one of the two equiv- alent conditions (a) and (b) in the following Lemma 1, then there exists a symmetric doubly stochastic matrix of order n with as its spectrum. T T Lemma 1. For a real n-tuple = (1,λ2,...,λn) . Let = (δ1,δ2,...,δn) . Then the following are equivalent. 1 λ λ λ (a) + 2 + 3 +···+ n 0, (2) n n(n − 1) (n − 1)(n − 2) 2 · 1 δ δ δ − (b) 1 + 2 +···+ n 1 + δ 0. (3) n n − 1 2 n Proof. We see that 1 + λ2 + λ3 +···+ λn − − − · n n(n 1) (n 1)(n 2) 2 1 1 1 1 1 1 = 1 − 0 + λ − +···+λ − n 2 n − 1 n n 1 2 1 1 1 = (1 − λ ) + (λ − λ ) +···+(λ − − λ ) + λ 2 n 2 3 n − 1 n 1 n 2 n δ δ δ − = 1 + 2 +···+ n 1 + δ . n n − 1 2 n Thus the equivalence of (a) and (b) follows. Notice that 1 λ λ 1 ←−− + 2 + 3 +···+ = Th , n n(n − 1) (n − 1)(n − 2) 2 · 1 n δ δ δ − ←− 1 + 2 +···+ n 1 + δ = ()Th . n n − 1 2 n n 80 S.-G. Hwang, S.-S. Pyo / Linear Algebra and its Applications 379 (2004) 77–83 Thus the inequalities (2) and (3) can be expressed as ←−− · hn 0, (4) ←− · hn 0(5) respectively, where ‘ · ’ stands for the Euclidean inner product. In the sequel we de- note by In,Jn,On the identity matrix, the all 1’s matrix and the zero matrix of order n respectively. We let en denote a column of Jn. We sometimes write I,J,O,e in place of In,Jn,On, en in case that the size of the matrix or vector is clear within the context. 2. Main result n , For 2 let 1 1 eT Qn = n n . (6) −e In−1 Then clearly Qn is nonsingular. We first observe the effect of the similarity transformation X → QXQ−1 on cer- tain sets of matrices on which our discussion relies. Lemma 2. Let Qn be the matrix defined in (6), then −1 = ⊕ (a)QnJnQn nI1 On−1, ⊕ −1 = ⊕ ∈ 1 (b)Qn(I1 A)Qn I1 A, for any A n−1. Proof. Clearly 1 eT n 0T Q J = = Q , n n 0 O 0 O n and (a) follows. To show (b), observe that 1 0T 1 1 eTA 1 1 eT Q = n n = n n , n 0 A −e A −e A 1 0T 1 1 eT 1 1 eT Q = n n = n n , 0 A n −Ae A −e A ∈ 1 for any A n−1. Thus (b) holds. s We first find a matrix with a prescribed spectrum in the set n−1. T Theorem 3. Let n 2. Then for any real n-tuple = (λ1,λ2,...,λn) , there λ1 exists a symmetric matrix A ∈ n with σ (A) = . S.-G. Hwang, S.-S. Pyo / Linear Algebra and its Applications 379 (2004) 77–83 81 Proof. Let (x1,x2,...,xn) be an n-tuple of indeterminates and let 1 1 1 A = x J + x I ⊕ J − +···+x I − ⊕ J . (7) 1 n n 2 1 n − 1 n 1 n n 1 1 1 We show that A is similar to diag(y1,y2,...,yn) where yi = xi + xi+1 +···+xn (i = 1, 2,...,n). We proceed by induction on n. If n = 2, then x x 1 − 1 1 1 + x 1 1 − x + x 0 Q AQ 1 = 2 2 2 2 2 2 = 1 2 , 2 2 − x1 x1 + x 1 0 x 11 2 2 2 1 2 2 and the induction starts. Suppose that n>2. We have, by Lemma 1, that −1 1 Q AQ = x I ⊕ O − + x I ⊕ J − n n 1 ( 1 n 1) 2 1 − n 1 n 1 1 1 + x I ⊕ J − +···+x I − ⊕ J 3 2 n − 2 n 2 n n 1 1 1 = y1I1 ⊕ B, where 1 1 1 B = x J − + x I ⊕ J − +···+x I − ⊕ J .
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