CONSTRUCTIONS OF TRACE ZERO SYMMETRIC STOCHASTIC MATRICES FOR THE INVERSE EIGENVALUE PROBLEM ROBERT REAMS

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1 CONSTRUCTIONS OF TRACE ZERO SYMMETRIC STOCHASTIC MATRICES FOR THE INVERSE EIGENVALUE PROBLEM ROBERT REAMS Abstract. I the special case of where the spectrum σ = {λ 1,λ 2,λ, 0, 0,...,0} has at most three ozero eigevalues λ 1,λ 2,λ with λ 1 0 λ 2 λ,adλ 1 + λ 2 + λ =0,theiverse eigevalue problem for symmetric stochastic matrices is solved. Costructios are provided for the appropriate matrices where they are readily available. It is show that whe is odd it is ot possible to realize the spectrum σ with a symmetric stochastic matrix whe λ 0ad 2 > λ 2 λ 0, adit is show that this boudis best possible. Key words. Iverse eigevalue problem, Symmetric stochastic matrix, Symmetric oegative matrix, Distace matrix. AMS subject classificatios. 15A18, 15A48, 15A51, 15A57 1. Itroductio. Let e 1,...,e deote the stadard basis i R,soe i deotes the vector with a 1 i the ith positio ad zeroes elsewhere. We will deote by e the vector of all oes, i.e. e =[1, 1,...,1] T R. A matrix A R is said to be stochastic whe all of its etries are oegative ad all its row sums are equal to 1, i.e. A has right eigevector e correspodig to the eigevalue 1. We will be cocered with symmetric stochastic matrices, so that these matrices are i fact doubly stochastic. Also, the eigevalues will ecessarily be real. If A R is oegative, has eigevalue λ 1 > 0 correspodig to the right eigevector e the 1 λ 1 A is stochastic, ad for coveiece we will state our results i the form, for example, of a matrix A havig eigevector e correspodig to 1 + ɛ, where the spectrum σ = {1 +ɛ, 1, ɛ, 0, 0,...,0}, with0 ɛ 1. We will say that σ = {λ 1,...,λ } R is realized as the spectrum of a matrix A i the evet that the matrix A has eigevalues λ 1,...,λ. The oegative iverse eigevalue problem is to fid ecessary ad sufficiet coditios that the elemets of the set σ = {λ 1,...,λ } C are the eigevalues of a matrix with oegative etries. This problem is curretly usolved except i special cases [1], [7], [8], [9], [10]. The restrictio of this problem to symmetric oegative matrices for which the eigevalues λ 1,...,λ satisfy λ 1 0 λ 2 λ is solved i [], where it is show that the oly ecessary ad sufficiet coditio is that i=1 λ i 0. Distace matrices are ecessarily symmetric, oegative, have trace zero, ad must have λ 1 0 λ 2 λ, although these are ot sufficiet coditios to be a distace matrix. We cojectured i [6], after solvig umerous special cases icludig =2,, 4, 5, 6, that the oly ecessary ad sufficiet coditios for the existece of a distace matrix with a give σ is that λ 1 0 λ 2 λ ad i=1 λ i = 0. Distace matrices with eigevector e were previously studied i [5], Receivedby the editors o 1 October Fial mauscript acceptedfor publicatio o 4 September Hadlig Editor: Tom Laffey. Departmet of Mathematics, Natioal Uiversity of IreladGalway, Galway, Irelad (robert.reams@uigalway.ie). 270

2 R. Reams 271 although ot i the cotext of these matrices eigevalues. Bouds o the eigevalues of symmetric stochastic matrices are give i [2] ad [4], although they do ot provide a restrictio o the eigevalues for the matrices uder cosideratio here. I Sectio 2 we provide a explicit costructio of a symmetric stochastic matrix which realizes the spectrum {2, 1, 1, 0, 0,...,0}, followed by showig that it is ot possible to realize the spectrum {1, 1, 0, 0,...,0} with a symmetric stochastic matrix whe is odd, although it is possible to realize this spectrum whe is eve. I Sectio we provide explicit costructios of symmetric stochastic matrices to realize {1 +ɛ, 1, ɛ, 0, 0,...,0}, for1 ɛ 0, whe is eve. We the show that it is ot possible to realize this spectrum with a symmetric stochastic matrix whe 2 >ɛ 0, ad is odd. Although we ca realize this spectrum with a symmetric stochastic matrix whe 1 ɛ 2,ad is odd. I the latter case we do ot provide a explicit costructio, istead makig use of the Itermediate Value Theorem i several variables. 2. Freedom ad restrictios o spectra. Lemma 2.1 will be used to establish that the matrix uder cosideratio is oegative. Lemma 2.1. Let A =(a ij ) R be a symmetric matrix with eigevalues λ 1,...,λ which satisfy λ 1 0 λ 2 λ. Suppose that A has eigevector e correspodig to λ 1, ad that A has all diagoal etries equal to zero. The A is a oegative matrix. ee Proof. Write A = λ T 1 + λ 2u 2 u T λ u u T,whereu 2,...,u are uit eigevectors correspodig to λ 2,...,λ, respectively. The 2a ij =(e i e j ) T A(e i e j )=λ 2 ((e i e j ) T u 2 ) λ ((e i e j ) T u ) 2 0, for all i, j, 1 i, j. Our ext two theorems give some foretaste of what is ad is t possible with the iverse eigevalue problem for symmetric oegative matrices. Note first that a symmetric oegative matrix with eigevector e ad of trace zero must be a oegative scalar multiple of A = ee T I, which has spectrum σ = {2, 1, 1}. Theorem 2.2. Let σ = {2, 1, 1, 0, 0,...,0}. The σ ca be realized as the spectrum of a symmetric oegative matrix A R with eigevector e correspodig to 2, for. Proof. Letu =[u 1,...,u ] T,v =[v 1,...,v ] T R 2 be give by u j = cos θ j, 2 v j = si θ j,whereθ j = 2π j,for1 j. The A =2eeT uut vv T is symmetric ad has zero diagoal etries by costructio. Sice the roots of x 1 are e 2πi j,1 j, the (coefficiet of x 1 )=0= 2πi j=1 e j ad (coefficiet of x 2 )=0= j,k=1,j k e 2πi j e 2πi k =( j=1 e 2πi j ) 2 j=1 e 2πi 2j. It follows that j=1 cos θ j = j=1 si θ j = j=1 si 2θ j = 0, which tells us that u T e = v T e = u T v = 0, respectively. It also follows that j=1 cos 2θ j =0,sothat j=1 cos2 θ j j=1 si2 θ j = 0. We also kow that j=1 cos2 θ j + j=1 si2 θ j =, soweca coclude that j=1 cos2 θ j = j=1 si2 θ j = 2. This tells us that u ad v are uit vectors. Thus A has spectrum σ ad is oegative from Lemma 2.1. Theorem 2.. Let λ 1 > 0 ad σ = {λ 1, λ 1, 0,...,0}. The σ caot be realized as the spectrum of a symmetric oegative matrix with eigevector

3 272 Trace Zero Symmetric Stochastic Matrices for the Iverse Eigevalue Problem e correspodig to λ 1,whe is odd. Proof. Suppose A is a matrix of the give form that realizes σ. WriteA = ee λ T 1 λ 1uu T, where u = [u 1,...,u ] T R, u = 1 ad u T e = 0. A has trace zero ad is oegative by hypothesis so all diagoal etries are zero, ad thus 0= λ1 λ 1u 2 i,for1 i. But the u i = ±1 ad 0 = i=1 u i = i=1 ±1,which is ot possible whe is odd. Remark 2.4. The same σ of Theorem 2. ca be realized by a symmetric oegative matrix with eigevector e correspodig to λ 1 whe is eve, by takig i the proof the uit vector u = 1 [1, 1,...,1, 1, 1,..., 1] T R.. At most three ozero eigevalues. The idea i Remark 2.4 ca be exteded to the case where is a multiple of 4 for the case of at most three ozero eigevalues. Theorem.1. Let σ = {1 +ɛ, 1, ɛ, 0, 0,...,0}, where 1 ɛ 0. The σ ca be realized by a symmetric oegative matrix A R with eigevector e correspodig to, whe =4m for some m. Proof. Letu, v R be give by u = 1 [1, 1,...,1, 1, 1,..., 1, 1, 1,..., 1, 1, 1,..., 1] T, v = 1 [1, 1,...,1, 1, 1,..., 1, 1, 1,..., 1, 1, 1,..., 1] T. The e, u ad v are orthoormal. Let A = () eet uu T ɛvv T, ad ote that all the diagoal etries of A are zero. However, the way we deal with the remaiig cases for eve is somewhat more complicated. Theorem.2. Let σ = {1 +ɛ, 1, ɛ, 0, 0,...,0}, where 1 ɛ 0. The σ ca be realized by a symmetric oegative matrix A R with eigevector e correspodig to, whe =4m +2 for some m. Proof. Let u =[u 1,...,u ] T, v =[v 1,...,v ] T R,adA =() eet uu T ɛvv T. We will require that 0 = u2 i ɛv2 i,for1 i, sothata has all zero diagoal etries. Let u i = cos θ i ad v i = ɛ si θ i,where the θ i s are chose below. Our θ i s must satisfy i=1 cos θ i = i=1 si θ i = i=1 cos θ i si θ i =0. Sothatu ad v are uit vectors we must have i=1 cos2 θ i = ad i=1 si2 θ i = ɛ, which will be achieved if i=1 cos2 θ i i=1 si2 θ i = (1 ɛ),ad i=1 cos2 θ i + i=1 si2 θ i =. So for ay give ɛ [0, 1] we also require for our choice of θ i s that i=1 cos 2θ i = (1 ɛ). Let 0 δ< 2π, ad choose the agles θ i,1 i, aroud the origi i the plae. Let θ i for 1 i m be, respectively, the 1st quadrat agles 2π δ, 2(2π δ),...,(m 1)( 2π δ),m( 2π δ). Let θ i for m +1 i 2m be, respectively, the 2d quadrat agles (m +1) 2π + mδ, (m +2)2π +(m 1)δ,...,(2m 1) 2π +2δ, 2m 2π + δ. Let θ 2m+1 = π. Let θ i for 2m +2 i m + 1 be, respectively, the rd quadrat agles 2m 2π δ, (2m 1) 2π 2δ,..., (m +2)2π (m 1)δ, (m +1)2π mδ.

4 R. Reams 27 Let θ i for m +2 i 4m + 1 be, respectively, the 4th quadrat agles m( 2π δ), (m 1)( 2π δ),..., 2( 2π δ), ( 2π δ). Let θ 4m+2 =0. For each θ i there is a θ i + π i the above list, by pairig off 1st quadrat agles with appropriate rd quadrat agles, ad pairig 2d quadrat agles with appropriate 4th quadrat agles. Therefore sice cos (θ i + π) = cos θ i we coclude that i=1 cos θ i =0. For each θ i 0 ad θ i π i the above list we ca pair off ay θ i with a correspodig θ i ad coclude that i=1 si θ i =0. Similarly, i=1 si 2θ i =0. Pairig off each θ i with a θ i + π i the same way as before, ad sice cos 2θ i = cos (2θ i +2π), we must have i=1 cos 2θ i =2+ 1st,2d,rd,4th quadrat cos 2θ i =2+2 1st,4th quadrat cos 2θ i. Because we ca pair off each θ i i the 1st quadrat with each θ i i the 4th quadrat i=1 cos 2θ i =2+4 1st quadrat cos 2θ i. Next usig the trigoometric formula j=0 cos jα = si ( +1 2 α)cos( 2 α) si α 2 (foud i [11], for example), where α =2( 2π δ) ad some simplificatio we obtai si ((2m +1)δ) that cos 2θ i =2 π si ( i=1 2m+1 δ). Let f(δ) =2 si ((2m+1)δ) π si ( 2m+1 δ),thef(0) = 0 ad lim f(δ) =, ad otice that f is cotiuous o the iterval [0, 2π δ 2π ). Also, sice 0 ɛ 1wehavethat0 (1 ɛ), but the by the Itermediate Value Theorem for each ɛ (0, 1] there is a δ [0, 2π (1 ɛ) )suchthatf(δ) =. The case ɛ =0ad δ = 2π is covered by the remark after Theorem 2.. I order to deal with the case where is odd we will improve o Theorem 2.. Theorem.. Let σ = {1 +ɛ, 1, ɛ, 0, 0,...,0}, where 2 >ɛ 0, ad. Theσ caot be realized by a symmetric oegative matrix A R with eigevector e correspodig to, whe =2m +1 for some m. Proof. σ is realizable whe ɛ = 1 from Theorem 2.2. We wish to determie the miimum value of ɛ for which it is possible to costruct a matrix of the desired form. Usig the same otatio as i the proof of Theorem.2 we wish to determie the miimum value of ɛ as a fuctio of θ 1,...,θ subject to the three costraits i=1 cos θ i = i=1 si θ i =0ad i=1 cos θ i si θ i = 0 (we kow from Theorem 2. that ɛ > 0). Observe that fidig the miimum ɛ is equivalet to fidig the maximum value of the fuctio = F (θ 1,...,θ )= i=1 cos2 θ i subject to the three costraits. For the momet let us determie the maximum value of F subject to the oe costrait i=1 cos θ i =0. Letλ deote the Lagrage multiplier so that 2 cosθ i si θ i λ si θ i = 0, i.e. si θ i (2 cos θ i + λ) = 0, for each i, 1 i. The

5 274 Trace Zero Symmetric Stochastic Matrices for the Iverse Eigevalue Problem for each i we have si θ i = 0 or else cos θ i = λ/2. We caot have all θ i s equal to0orπ, sice the we would ot have i=1 cos θ i = 0, because is odd. Suppose ow that k of the cos θ i s are equal, but ot equal to ±1, the k cos θ i = ±1 ± 1. The i order to maximize F we must have as may ±1 s as possible, thus we must have k =2withcosθ p =cosθ q (say) ad the remaiig θ i s all either 0 or π. The 2cosθ i =1or 1, ad there are two possibilities: Case 1: cos θ p =cosθ q = 1 2 with m 1 of the remaiig cos θ i sequalto1adthe other m of the cos θ i s equal to 1. Case 2: cos θ p =cosθ q = 1 2 with m 1 of the remaiig cos θ i s equal to 1 ad the other m of the cos θ i s equal to 1. I either case the maximum value of F is m 1= 2,iwhichcase ɛ = 2. Notice ow that this maximum value for F ca be just as easily achieved whe θ p = θ q ad that the i=1 si θ i =0ad i=1 cos θ i si θ i =0. Soɛ has i effect bee miimized subject to all three costraits. Corollary.4. Let σ = {1 +ɛ, 1, ɛ, 0,...,0}, where1 ɛ 2,ad. The σ ca realized as the spectrum of a symmetric oegative matrix A R with eigevector e correspodig to, whe is odd. Proof. LetF (θ 1,...,θ )= j=1 cos2 θ j.the ad F ( 2π 1, 2π 2,...,2π ( 1), 2π )= 2, F ( 2π, 2π, 0, 0,...,0,π,π,...,π)= 2. Moreover, F is cotiuous as a fuctio of cos θ 1,...,cos θ,particularlyothe followig itervals for the cos θ i s: For = 4k + let the cos θ i s satisfy cos 2π(k+1) cos θ k+1 cos 2π ad cos 2π(k+) cos θ k+ cos 2π. Also, let cos 2π j cos θ j cos 0 for 1 j k ad k +4 j 4k +,adletcosπ cos θ j cos 2π j for k +2 j k + 2 except that cos θ 2k+2 =cosπ. For = 4k +1 let the cos θ i s satisfy cos 2π(k+1) cos θ k+1 cos 2π ad cos 2π(k+2) cos θ k+2 cos 2π. Also, let cos 2π j cos θ j cos 0 for 1 j k ad k + j 4k +1,adletcosπ cos θ j cos 2π j for k +2 j k + 1 except that cos θ 1 = cos 0. For each ɛ such that 1 ɛ 2 we have 2 2, the from the Itermediate Value Theorem for real valued fuctios of several variables it follows that for each ɛ [ 2, 1] there is a (θ 1,...,θ )suchthatf(θ 1,...,θ )=. The author does ot see a atural extesio of the above methods to deal with the case of at most four ozero eigevalues. Ackowledgemet. I am grateful to Ro Smith ad Charles R. Johso for some iterestig coversatios.

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