c 2006 Society for Industrial and Applied Mathematics

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1 SIAM J. MATRIX ANAL. APPL. Vol. 7, No. 3, pp c 006 Society for Idustrial ad Applied Mathematics EXTREMAL EIGENVALUES OF REAL SYMMETRIC MATRICES WITH ENTRIES IN AN INTERVAL XINGZHI ZHAN Abstract. We determie the exact rage of the smallest ad largest eigevalues of real symmetric matrices of a give order whose etries are i a give iterval. The maximizig ad miimizig matrices are specified. We also cosider the maximal spread of such matrices. Key words. eigevalue, symmetric matrix, spread AMS subject classificatios. 15A18, 15A4, 15A57 DOI / Itroductio. Let S a, b deote the set of real symmetric matrices whose etries are i the iterval a, b. For a real symmetric matrix A, we always deote the eigevalues of A i decreasig order by λ 1 A) λ A). We will study the smallest eigevalue λ A) ad the largest eigevalue λ 1 A) whe A varies i S a, b. Costatie proved that if A S 0,b, the { b/ if is eve, λ A) 1b/ if is odd. So the matrices treated there are oegative. The proof techiques of are graphtheoretic. I 7 Roth gave aother proof of this result by aalysis of eigevectors. I this paper we will determie the smallest ad largest values of both λ A) ad λ 1 A) whe A S a, b for geeric a<b, thus geeralizig Costatie s result. The spread of a real symmetric matrix A is defied as sa) =λ 1 A) λ A). This quatity has applicatios i combiatorial optimizatio problems 3. Some lower bouds o the spread of Hermitia matrices are kow; see 5 ad the refereces therei. We will determie the maximal value of sa) for A S a, a. We always regard real vectors i R as 1 matrices. A basic fact see 1 or 4) for a real symmetric matrix A is λ A) = mi{x T Ax : x =1,x R }, λ 1 A) = max{x T Ax : x =1,x R }.. Extremal eigevalues. We first cosider the lower boud for the smallest eigevalue. Deote by J r,s the r s matrix with all etries equal to 1, ad write J r for J r,r. Theorem 1. Let A S a, b with ad a<b. i) If a <b, the { a b)/ if is eve, λ A) a ) a + 1)b / if is odd. Received by the editors March 7, 005; accepted for publicatio i revised form) by R. Bhatia July 6, 005; published electroically Jauary 7, 006. This work was supported by NSFC grat Departmet of Mathematics, East Chia Normal Uiversity, Shaghai 0006, Chia zha@ math.ecu.edu.c). 851

2 85 XINGZHI ZHAN If is eve, equality holds if ad oly if A is permutatio similar to a b J b a. If is odd, equality holds if ad oly if A is permutatio similar to aj 1 bj 1, +1. bj +1, 1 aj +1 ii) If a b, the λ A) a. If a >b, equality holds if ad oly if A = aj. If a = b, equality holds if ad oly if A is permutatio similar to aj k bj k, k bj k,k aj k for some k with 1 k. Proof. For ay fixed A S a, b, let x =x 1,...,x ) T be a uit i.e., x =1) eigevector correspodig to λ A). By simultaeous permutatios of the rows ad colums of A if ecessary, we may suppose that x i 0 for i =1,...,k ad x j < 0 for j = k +1,...,,1 k. We eed ot cosider the case k = 0, as i that case we use x istead of x. Let e R be the vector with all etries equal to 1. Deote by A B the Hadamard product of A ad B, i.e., the etrywise product. The we may write x T Ax i a more visible form: 1) λ A) =x T Ax = e T A xx T )e. Note that the matrix xx T is divided ito four blocks: The etries i xx T )1,...,k ad i xx T )k +1,..., are oegative, while the etries i xx T )1,...,k k +1,..., ad i xx T )k +1,..., 1,...,k are opositive. Thus from 1) we see clearly that if we defie aj à = Jk; a, b) k bj ) k, k, the bj k,k aj k λ Ã) = mi{yt Ãy : y =1,y R } x T Ãx x T Ax = λ A). Therefore the smallest value of λ A) for A S a, b ca be attaied at some matrix of the form i ). Sice the rak of Jk; a, b) is at most, it has at most two ozero eigevalues. By cosiderig the trace ad the Frobeius orm we deduce that λ Jk; a, b)) = a ) 3) k) a +4k k)b /. i) Sice a <b,if is eve, the right side of 3) attais its miimum at k = / ad λ A) λ J/; a, b)) = a b)/ for ay A S a, b. If is odd, the right side of 3) attais its miimum at k = 1)/ ad k = +1)/. Hece λ A) λ J 1)/; a, b)) = a ) a + 1)b /.

3 EIGENVALUES OF REAL SYMMETRIC MATRICES 853 Now we prove the equality coditios. First suppose is eve. Let A =a ij ) S a, b such that λ A) =a b)/ ad let x =x 1,...,x ) T be a correspodig uit eigevector. Suppose x has exactly t ozero compoets with t <. By the above bouds, if t is eve, the ad if t is odd, the λ A) =x T Ax λ A) =x T Ax ta b)/ >a b)/, ta a b )+t b ) / >ta b)/ >a b)/, both cotradictig the assumptio that λ A) =a b)/. Therefore all the compoets of x are ozero. Suppose x has k positive compoets ad k egative compoets. From 3) we kow that if k /, λ A) λ Jk; a, b)) >λ J/; a, b)) = a b)/, a cotradictio. Thus we must have k = /. By simultaeous row ad colum permutatios of A if ecessary, we may suppose that x i > 0 for i =1,...,/ ad x j < 0 for j =/)+1,...,. The λ A) =x T Ax x T J/; a, b)x λ A) forces A = J/; a, b), sice otherwise the first iequality above will be strict, which is impossible. Therefore the origial A is permutatio similar to J/; a, b). The equality coditio for the case whe is odd ca be similarly proved. Just ote that ta b) >a a + 1)b for 1 t 1, the lower boud a a + 1)b )/ is strictly decreasig i, ad aj +1 bj +1, 1 aj 1 bj 1 ad, +1 bj 1, +1 aj 1 bj +1, 1 aj +1 are permutatio similar. ii) a b ad a<bimply a<0. If a >b, the miimum of the right side of 3) is attaied at k =, ad if a = b, the right side of 3) has equal values for all k with 1 k. I ay case the miimum is a. This proves λ A) a. The proof of the equality coditios is similar to that of case i) ad we omit the details. Sice A S a, b implies A aj S 0,b a, it is atural to ask whether Theorem 1 ca be deduced from Costatie s result by usig the perturbatio iequality: λ G + H) λ G)+λ H) for real symmetric matrices G, H 1. I geeral the aswer is o. Let us examie the case 0 <a<b. If is odd, the perturbatio iequality ad Costatie s result give λ A) =λ A aj )+aj λ A aj )+λ aj ) = λ A aj ) 1a b)/.

4 854 XINGZHI ZHAN It is easy to verify that this lower boud 1a b)/ is strictly less tha the sharp boud a a + 1)b )/ i Theorem 1. O the other had, if is eve, the lower boud a b)/ ca ideed be deduced from Costatie s result. For a real symmetric matrix A, λ 1 A) = λ A). Also a a ij b is equivalet to b a ij a. Thus the followig corollary o upper bouds for the largest eigevalue follows from Theorem 1. Corollary. Let A S a, b with ad a<b. i) If a< b, the { b a)/ if is eve, λ 1 A) b + ) b + 1)a / if is odd. If is eve, equality holds if ad oly if A is permutatio similar to b a J a b. If is odd, equality holds if ad oly if A is permutatio similar to bj 1 aj 1, +1. aj +1, 1 bj +1 ii) If a b, the λ 1 A) b. If a> b, equality holds if ad oly if A = bj.ifa = b, equality holds if ad oly if A is permutatio similar to bj k aj k, k aj k,k bj k for some k with 1 k. Now we tur to the study of upper bouds o the smallest eigevalue ad lower bouds o the largest eigevalue. For real matrices A, B, we write A B to mea that B A is etrywise oegative. We eed the followig two lemmas. Lemma 3 see 1 or 4). Let H be a real symmetric matrix of order ad G be a pricipal submatrix of order k of H. The λ j H) λ j G) λ j+ k H) for j =1,...,k. Lemma 4 see 6, p. 38). Let A, B be oegative matrices of the same order satisfyig A B. The ρa) ρb), where ρ ) is the Perro root spectral radius). If, i additio, A B ad B is irreducible, the ρa) <ρb). Sice λ A) = λ 1 A) ad λ 1 A) = λ A) for real symmetric matrices A, for our problem there are essetially two differet cases: 0 <a<bad a 0 <b. Deote by J ad I the all-oe matrix ad the idetity matrix, respectively. Theorem 5. Let A S a, b with ad a<b. i) Let 0 <a<b. The 4) λ A) b a. Equality i 4) holds if ad oly if A = aj +b a)i. 5) λ 1 A) a. Equality i 5) holds if ad oly if A = aj.

5 ii) Let a 0 <b. The EIGENVALUES OF REAL SYMMETRIC MATRICES 855 6) λ A) b. Equality i 6) holds if ad oly if A = bi. 7) λ 1 A) a. Equality i 7) holds if ad oly if A = ai. Proof. i) Let A =a ij ). For i<j, by Lemma 3 we have 8) λ A) λ aii a ji a ij a jj ) = a ii + a jj a ii a jj ) +4a ij a ii + a jj a ij b a. Thus λ A) b a. If λ A) =b a, all the iequalities i 8) must be equality. This forces a ii = a jj = b ad a ij = a ji = a. As this should be true for all i<j, A = aj +b a)i. A S a, b implies A aj 0. By Lemma 4 ad the Perro Frobeius theory 6, λ 1 A) =ρa) ρaj) =λ 1 aj) =a. If A S a, b ad A aj, the sice A is irreducible A is i fact etrywise positive), agai by Lemma 4, λ 1 A) >λ 1 aj) =a. Thusλ 1 A) =a if ad oly if A = aj. ii) λ A) tra b = b. If λ A) =b, the tra = b ad cosequetly a ii = b, i =1,...,. For ay i<j,by Lemma 3 we have b aij b = λ A) λ a ji b Thus a ij = 0 for all i<j, i.e., A = bi. λ 1 A) tra ) = b a ij. a = a. If λ 1 A) =a, the tra = a ad hece a ii = a, i =1,...,. For ay i<j,by Lemma 3 we have a aij a = λ 1 A) λ 1 a ji a ) = a + a ij. So a ij = 0 for all i<j, i.e., A = ai. This completes the proof.

6 856 XINGZHI ZHAN 3. The maximal spread. Deote by sa) the spread of A. We treat oly the case whe the iterval is symmetric about the origi. Of course we may use the upper boud o λ 1 i Corollary ad the lower boud o λ i Theorem 1 to give a upper boud o the spread λ 1 λ, but that boud is ot sharp. This is because the upper boud o λ 1 ad the lower boud o λ caot be simultaeously attaied at oe commo matrix. A {±1}-matrix is a matrix whose etries are either 1 or 1. Two matrices A, B of the same order are said to be D-similar if there is a diagoal matrix D with diagoal etries equal to 1 or 1 such that DAD = B. We will eed the followig lemma. Lemma 6. Let A be a symmetric {±1}-matrix with all diagoal etries equal to 1. The either A is D-similar to J or A has a pricipal submatrix of order 3 which is similar to B 1 = Proof. The cases =1, are obvious. Use iductio o for 3. First let =3. IfA = J 3, there is othig to prove. Otherwise A has a off-diagoal etry equal to 1. The there are the followig possibilities of A: B 1,, 1 1 1, 1 1 1, 1 1 1, , It is easy to check that each of these matrices satisfies the coclusio. Now cosider 4 ad suppose the lemma holds for matrices of order 1. Let 1 u T A =. u A 1 If A 1 has a pricipal submatrix of order 3 which is similar to B 1, the so does A. Otherwise, by the assumptio, A 1 is D-similar to J 1.SoAis D-similar to 1 v T H = =h v J ij ). 1 If each etry of v is 1 or each etry of v is 1, the H ad hece A are D-similar to J. Otherwise we have h 1,p = 1, h 1,q =1orh 1,p =1,h 1,q = 1 for some 1 <p<q. I the first case H1,p,q=B 1. I the secod case H1,p,q is D-similar to B 1. Therefore i both cases A has a pricipal submatrix which is similar to B 1. Two matrices A, B of the same order are said to be sig-permutatio similar if there exist a permutatio matrix P ad a diagoal matrix D with diagoal etries equal to 1 or 1 such that DP T AP D = B. It is clear that sig-permutatio similarity is a equivalece relatio. Theorem 7. Let A S a, a with ad a>0. The { a if is eve, sa) 1a if is odd.

7 EIGENVALUES OF REAL SYMMETRIC MATRICES 857 If is eve, equality holds if ad oly if A is sig-permutatio similar to 1 1 a J 1 1. If is odd, equality holds if ad oly if A is sig-permutatio similar to J +1 J +1 ±a, 1. J 1, +1 J 1 Proof. By cosiderig a 1 A istead of A, it suffices to prove the theorem for the case a = 1. Suppose A S 1, 1. Throughout this proof we write λ j for λ j A). For x R, we always write its compoets as x 1,...,x. Give A S 1, 1, let x, y R be uit eigevectors such that λ 1 = x T Ax, λ = y T Ay. The 9) sa) =x T Ax y T Ay = e T A xx T yy T )e. Note that the i, j) etry of xx T yy T is x i x j y i y j. We defie a ew matrix à =ã ij) as ã ij =1ifx i x j y i y j 0 ad ã ij = 1 ifx i x j y i y j < 0. The from 9) we have sa) x T Ãx y T Ãy max{w T Ãw : w =1,w R } mi{z T Ãz : z =1,z R } = sã). Therefore the maximal spread ca always be attaied at some {±1}-matrix. If =, the coclusios of the theorem are easily checked to be true. Next we assume 3. Now suppose A is a {±1}-matrix of order. The followig three matrices will play a role i our proof: B 1 = Their eigevalues are, B = 1 1 1, B 3 = λb 1 )={,, 1}, λb )={, 1, }, λb 3 )={, 1, }. If A has a pricipal submatrix of order 3 which is similar to B 1, the by Lemma 3 λ. If A has a pricipal submatrix of order 3 which is similar to B, the by Lemma 3 λ 1. If A has a pricipal submatrix of order 3 which is similar to B 3, the by Lemma 3 λ 1 1. I all three cases we have 1 λ 1 + λ = A F λ j 1. j=. Hece 10) sa) =λ 1 λ λ 1 + λ ) { } 1) < mi, 1. Thus if oe of the above cases occurs, the spread is less tha our claimed upper boud.

8 858 XINGZHI ZHAN If all the diagoal etries of A are 1, the by Lemma 6 either A is D-similar to J or A has a pricipal submatrix of order 3 which is similar to B 1. Sice sj )=, i both cases sa) is ot the maximal value. If all the diagoal etries of A are 1, the sice s A) =sa), this case is the same as what we just discussed. Next cosider those {±1}-matrices A whose diagoal cotais both 1 ad 1. By simultaeous row ad colum permutatios if ecessary, we may suppose Ar A A = r,s A T, r,s A s where A r is of order r 1 r 1) ad A r s diagoal etries are all 1, ad A s is of order s = r ad A s s diagoal etries are all 1. Sice s A) =sa), we eed cosider oly the case r s. The r. By Lemma 6, either A r is D-similar to J r or A r has a pricipal submatrix of order 3 which is similar to B 1. I the first case A is D-similar to a matrix whose rth leadig pricipal submatrix is J r, while i the secod case sa) is ot the maximal value. Thus we may suppose A r = J r. Now two cases ca occur. i) A r,s has a colum which cotais both 1 ad 1. The A has a pricipal submatrix which is similar to B, ad sa) is ot the maximal value. ii) Each colum of A r,s cotais oly 1 or oly 1. The A is D-similar to Jr J r,s G =, J s,r where all the diagoal etries of Ãs remai 1. If Ãs has a off-diagoal etry equal to 1, the G has B 3 as a pricipal submatrix ad sg) is ot the maximal value. Thus we further cosider the case Ãs = J s. Now there remais the case Jr J 11) A = r,s, J s,r J s à s where r s. It is easy to see that Jr J 1) s r,s J s,r J s ) = r ). Therefore by 1) the maximal spread of the matrices i 11) is attaied uiquely at r = / if is eve, ad the maximal spread is 1 attaied uiquely at r = +1)/ if is odd. For the odd case ote that we have assumed r s = r, ad hece r = 1)/ does ot occur. At this stage we have foud the maximal spread of A S 1, 1. Next we determie those matrices which attai the maximal spread. Suppose A S 1, 1 attais the maximal spread ad x, y are the uit eigevectors correspodig to λ 1 ad λ, respectively. Assume that xx T yy T has a zero etry, say, x i x j y i y j =0 for some i, j. From 9) it is clear that we ca chage the correspodig etry a ij of A arbitrarily without affectig the value of sa). So we may suppose a ij = 0. The 1 A F λ 1 + λ λ 1 λ ) { if is eve, = 1 if is odd, which is a cotradictio. Thus every etry of xx T yy T is ozero. By 9) we deduce that A must be a {±1}-matrix. O the other had, the above aalysis leadig

9 EIGENVALUES OF REAL SYMMETRIC MATRICES 859 to the maximal spread shows that whe is eve, sa) = if ad oly if A is sig-permutatio similar to 1 1 J 1 1, ad whe is odd, sa) = 1 if ad oly if A is sig-permutatio similar to J +1 J +1 ±, 1. J 1, +1 J 1 The permutatio similarity comes from our operatio to put positive diagoal etries together ad let them appear first. The possible mius sig comes from the fact that s A) = sa), which we used to simplify our aalysis. Note also that i the case whe is eve, J 1 1 ad 1 1 J are sig-permutatio similar, ad the mius sig eed ot appear i our assertio for the equality case. This completes the proof. Let e t R t deote the all-oe vector. By Theorem 7 ad 9) we get the followig iterestig corollary. Corollary 8. max x i x j y i y j : x = y =1,x,y R = i,j=1 { if is eve, 1 if is odd. The maximum is attaied at x, y if ad oly if x = DPx 0, y = DPy 0 for some diagoal matrix D with diagoal etries equal to 1 or 1 ad some permutatio matrix P where ) ) ae/ be/ x 0 =, y be 0 =, a =1+ 1 )b, b = / ae / + ) if is eve ad x 0 = y 0 = ae+1)/ be 1)/ ce+1)/ de 1)/ ),a= ),c= ),b= 1 ),d= ) 1 ) 1 if is odd. We remark that the above x 0 ad y 0 are the uit eigevectors correspodig to the largest ad smallest eigevalues of the maximizig matrix i Theorem 7. We have the followig two obvious problems which are ot solved here. Problem 1. For a give iteger j with j 1, determie max{λ j A) :A S a, b}, mi{λ j A) :A S a, b}

10 860 XINGZHI ZHAN ad determie which matrices attai the maximum ad which matrices attai the miimum. Problem. For geeric a<b, determie max{sa) :A S a, b}, where sa) deotes the spread of A, ad determie which matrices attai the maximum. REFERENCES 1 R. Bhatia, Matrix Aalysis, Spriger-Verlag, New York, G. Costatie, Lower bouds o the spectra of symmetric matrices with oegative etries, Liear Algebra Appl., ), pp G. Fike, R. E. Burkard, ad F. Redl, Quadratic assigmet problems, i Surveys of Combiatorial Optimizatio, North Hollad, Amsterdam, 1987, pp R. A. Hor ad C. R. Johso, Matrix Aalysis, Cambridge Uiversity Press, Cambridge, UK, E. Jiag ad X. Zha, Lower bouds for the spread of a Hermitia matrix, Liear Algebra Appl., ), pp H. Mic, Noegative Matrices, Joh Wiley & Sos, New York, R. Roth, O the eigevectors belogig to the miimum eigevalue of a essetially oegative symmetric matrix with bipartite graph, Liear Algebra Appl., ), pp