The inverse eigenvalue problem for symmetric doubly stochastic matrices

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1 Liear Algebra ad its Applicatios 379 (004) The iverse eigevalue problem for symmetric doubly stochastic matrices Suk-Geu Hwag a,,, Sug-Soo Pyo b, a Departmet of Mathematics Educatio, Kyugpook Natioal Uiversity, Taegu 70-70, Republic of Korea b School of Electrical Egieerig ad Computer Sciece, Kyugpook Natioal Uiversity, Taegu 70-70, Republic of Korea Received July 00; accepted 4 December 00 Submitted by F. Uhlig Abstract For a positive iteger ad for a real umber s, let Γ s deote the set of all real matrices whose rows ad colums have sum s. I this ote, by a explicit costructive method, we prove the followig. (i) Give ay real -tuple Λ = (λ,λ,...,λ ) T, there exists a symmetric matrix i Γ λ whose spectrum is Λ. (ii) For a real -tuple Λ = (,λ,...,λ ) T with λ λ, if + λ ( ) + λ 3 ( ) + + λ 0, the there exists a symmetric doubly stochastic matrix whose spectrum is Λ. The secod assertio eables us to show that for ay λ,...,λ [ /( ), ], there is a symmetric doubly stochastic matrix whose spectrum is (,λ,...,λ ) T ad also that ay umber β (, ] is a eigevalue of a symmetric positive doubly stochastic matrix of ay order. 003 Elsevier Ic. All rights reserved. Correspodig author. Tel.: ; fax: addresses: sghwag@ku.ac.kr (S.-G. Hwag), sspyo@gauss.ku.ac.kr (S.-S. Pyo). Supported by Com MaC-KOSEF. This work was doe while the secod author had a BK Post-Doc. positio at Kyugpook Natioal Uiversity /$ - see frot matter 003 Elsevier Ic. All rights reserved. doi:0.06/s (03)

2 78 S.-G. Hwag, S.-S. Pyo / Liear Algebra ad its Applicatios 379 (004) AMS classificatio: 5A5; 5A8 Keywords: Doubly stochastic matrix; Iverse eigevalue problem; Spectrum. Itroductio A real matrix A is called oegative (resp. positive), writte A O (resp. A> O), if all of its etries are oegative (resp. positive). A square oegative matrix is called doubly stochastic if all of its rows ad colums have sum. The set of all doubly stochastic matrices is deoted by Ω. For a square matrix A, let σ (A) deote the spectrum of A. Give a -tuple Λ = (λ,λ,...,λ ) T of umbers, real or complex, decidig the existece of a matrix A with some specific properties such that σ (A) = Λ has log time bee oe of the problems of mai iterest i the theory of matrices. Give Λ = (λ,λ,...,λ ) T, i order that Λ = σ (A) for some positive matrix A, it is ecessary that λ + λ + +λ > 0. Sufficiet coditios for the existece of a positive matrix A with σ (A) = Λ have bee ivestigated by several authors such as Borobia [], Fiedler [], Kellog [4], ad Salzma [5]. I this paper we deal with the existece of certai real symmetric matrices ad that of symmetric doubly stochastic matrices with prescribed spectrum uder certai coditios. Let A Ω. The the well kow Gershgori s theorem (see [3], for istace) implies that σ (A) is cotaied i the closed uit disc cetered at the origi i the complex plae. Thus if, i additio, A is symmetric, the σ (A) lies i the real closed iterval [, ]. Give a real -tuple Λ = (λ,λ,...,λ ) T with λ λ λ, i order that there exists A Ω with σ (A) = Λ, it is ecessary that λ =, λ, λ + λ + +λ 0. () Thus, for istace, there exists o 3 3 doubly stochastic matrix with spectrum (, 0.5, 0.6) T or (,,.) T. Certaily the coditio () is ot sufficiet for Λ = (λ,λ,...,λ ) T with λ λ λ to be the spectrum of a doubly stochastic matrix. I this paper, we fid a sufficiet coditio o Λ = (λ,λ,...,λ ) T with λ λ λ satisfyig () for the existece of a symmetric doubly stochastic matrix havig Λ as its spectrum by a costructive method. For a -tuple x = (x, x,...,x ) T, let x ad x be defied by x = (x x,x x 3,...,x x,x ) T, x = (x,x,...,x,x ) T.

3 S.-G. Hwag, S.-S. Pyo / Liear Algebra ad its Applicatios 379 (004) Note that x is the differece sequece of the fiite sequece 0, x,x,..., x,x. Let h = (,, 3,..., ) T. The ( h =, 3,..., ( ), ) T. We see that the ith compoet of h is the legth of the ith subiterval of the divisio of the uit iterval [0, ] by isertig the poits, 3,...,. For a real umber s, let Γ s deote the set of all real matrices all of whose rows ad colums have sum s. The set Ω cosists of all oegative matrices i Γ. I what follows we show that for ay real -tuple Λ = (λ,λ,...,λ ) T, there exists a symmetric matrix i Γ λ whose spectrum is Λ, ad also that if a real -tuple Λ = (,λ,...,λ ) T with λ λ satisfies () ad oe of the two equivalet coditios (a) ad (b) i the followig Lemma, the there exists a symmetric doubly stochastic matrix of order with Λ as its spectrum. Lemma. For a real -tuple Λ = (,λ,...,λ ) T. Let Λ = (δ,δ,...,δ ) T. The the followig are equivalet. (a) + λ ( ) + λ 3 ( )( ) + + λ 0, () (b) δ + δ + + δ + δ 0. (3) Proof. We see that + λ ( ) + λ 3 ( )( ) + + λ ( ) ( = 0 + λ ) ( + +λ ) = ( λ ) + (λ λ 3 ) + +(λ λ ) + λ = δ + δ + + δ + δ. Thus the equivalece of (a) ad (b) follows. Notice that + λ ( ) + λ 3 ( )( ) + + = ΛT h, δ + δ + + δ + δ = ( Λ) T h.

4 80 S.-G. Hwag, S.-S. Pyo / Liear Algebra ad its Applicatios 379 (004) Thus the iequalities () ad (3) ca be expressed as Λ h 0, (4) Λ h 0 (5) respectively, where stads for the Euclidea ier product. I the sequel we deote by I,J,O the idetity matrix, the all s matrix ad the zero matrix of order respectively. We let e deote a colum of J. We sometimes write I,J,O,e i place of I,J,O, e i case that the size of the matrix or vector is clear withi the cotext.. Mai result For, let [ Q = e T ]. (6) e I The clearly Q is osigular. We first observe the effect of the similarity trasformatio X QXQ o certai sets of matrices o which our discussio relies. Lemma. Let Q be the matrix defied i (6), the (a)q J Q = I O, (b)q (I A)Q = I A, for ay A Γ. Proof. Clearly [ ] [ ] e T 0 T Q J = = Q 0 O 0 O, ad (a) follows. To show (b), observe that [ ] [ 0 T Q = e T ] [ A = e T 0 A e A e A [ ] [ 0 T Q 0 A = e T ] [ = e T Ae A e A ], ], for ay A Γ. Thus (b) holds. We first fid a matrix with a prescribed spectrum i the set Γ s. Theorem 3. Let. The for ay real -tuple Λ = (λ,λ,...,λ ) T, there exists a symmetric matrix A Γ λ with σ (A) = Λ.

5 S.-G. Hwag, S.-S. Pyo / Liear Algebra ad its Applicatios 379 (004) Proof. Let (x,x,...,x ) be a -tuple of idetermiates ad let ( ) ( A = x J + x I ) ( J + +x I ) J. (7) We show that A is similar to diag(y,y,...,y ) where y i = x i + x i+ + +x (i =,,...,). We proceed by iductio o. If =, the [ ][ Q AQ x = + x x ][ ] [ ] x + x x x = 0, + x 0 x ad the iductio starts. Suppose that >. We have, by Lemma, that ( Q AQ = x (I O ) + x I + x 3 ( I = y I B, J ) ( J + +x I ) J where ( ) ( B = x J + x 3 I ) ( J + +x I ) J. By the iductio hypothesis, there exists a osigular matrix Q of order such that QBQ = diag(y,y 3,...,y ). The (I Q)Q AQ (I Q ) = diag(y,y,...,y ). Now, let (x,x,...,x ) T = Λ. The (y,y,...,y ) T = (λ,λ,...,λ ) T, ad the theorem is proved. We are ow ready to prove the followig. Theorem 4. Let Λ = (,λ,...,λ ) T be a real -tuple with λ λ. If Λ satisfies oe of the iequalities () (5), the there exists a symmetric A Ω with σ (A) = Λ. Moreover, if >λ, the the matrix A ca be take to be positive. Proof. Let Λ = (δ,δ,...,δ ) T, ad let A =[a ij ] be the matrix defied as (7) with x i = δ i (i =,,...,). The A Λ. All we eed to show is that A O. Sice δ i 0fori =,,...,, we see that all of the off-diagoal etries of A are oegative. We also see, for each i =,,...,, that a ii = δ + δ + + δ i i + + δ i+ + +δ, )

6 8 S.-G. Hwag, S.-S. Pyo / Liear Algebra ad its Applicatios 379 (004) ad a = δ + δ + + δ + δ from which it follows that a a a because δ i 0fori =,,...,. Now that a 0 follows from Lemma. If >λ, the δ > 0, aditmustbethata>o. Theorem 4 ca be iterpreted geometrically as Corollary 5. Let Λ = (,λ,...,λ ) T be a real -tuple with λ λ. If Λ forms a acute or right agle with h, the there exists a symmetric A Ω with σ (A) = Λ. Let be fixed ad let Π 0 deote the hyperplae {x R x T h = 0}, where R deotes the real Euclidea -space. The R Π 0 is divided ito two parts Π + ={x R x T h > 0}, Π ={x R x T h < 0}. Let S ={(,x,...,x ) T R x x }. The Corollary 5 ca be restated as Corollary 6. Every vector i S (Π + Π 0 ) is the spectrum of a symmetric doubly stochastic matrix. A matrix is irreducible if there does ot exist a permutatio matrix P such that P T AP has the form [ ] P T B O AP =, C where B,C are ovacuous square matrices. If A is a doubly stochastic, the there are irreducible doubly stochastic matrices A,A,...,A k, called the irreducible compoets of A, such that A = A A A k. The algebraic multiplicity of the eigevalue of A equals the umber of irreducible compoets of A. For Λ = (,λ,...,λ ) T, with λ λ, if = λ, the there is o irreducible doubly stochastic matrix whose spectrum is Λ. But if >λ, the the matrix defied by (7) with x i = δ i,i=,,...,,is a positive matrix. From the proof of Theorem 3, we get a δ + δ + +δ + λ = + ( )λ.

7 S.-G. Hwag, S.-S. Pyo / Liear Algebra ad its Applicatios 379 (004) Thus we have Corollary 7. If λ,...,λ [ /( ), ], the there exists a symmetric A Ω with σ (A) = (,λ,...,λ ) T. Note that the lower boud /( ) of the iterval is the best possible for the statemet of Corollary 7. For, if α is a umber such that α< /( ), the there does ot exist a A Ω such that σ (A) = (,α,...,α) because A must have a oegative trace. Let β be ay umber such that <β<. Take ay real umber ε such that 0 <ε<mi{ + β, β}. Let {, if i =, λ i = ε, if i, β, if i = ad let Λ = (λ,λ,...,λ ) T. The Λ = (δ,δ,...,δ ) T = (ε, 0,...,0, ε β,β) T so that δ + δ + + δ + δ = ε + ε β + β> + β ε > 0. Thus we have the followig. Corollary 8. Let, the for ay β (, ], there exists a symmetric positive A Ω of which β is a eigevalue. Certaily is a eigevalue of some A Ω for every, for example, of [ ] 0 I 0. But caot be a eigevalue of a positive doubly stochastic matrix. This fact follows directly from the Perro Frobeius theory. We give here aother simple proof. Suppose that Ax = x where A =[a ij ] Ω,A>O,ad x = (x,x,...,x ) T. Without loss of geerality we may assume that x = ad x i foralli =, 3,...,.The a x + a 3 x 3 + +a x = a from which we have < + a <a x + +a x a + +a = a <, a cotradictio. Refereces [] A. Borobia, O the oegative eigevalue problem, Liear Algebra Appl. 3/4 (995) [] M. Fiedler, Eigevalues of oegative symmetric matrices, Liear Algebra Appl. 9 (974) 9 4. [3] R. Hor, C. Johso, Matrix Aalysis, Cambridge Uiversity Press, 985. [4] R.B. Kellog, Matrices similar to a positive or essetially positive matix, Liear Algebra Appl. 4 (97) [5] F.L. Salzma, A ote o eigevalues of oegative matrices, Liear Algebra Appl. 5 (97)