On the construction of symmetric nonnegative matrix with prescribed Ritz values

Transcription

1 Joural of Lear ad Topologcal Algebra Vol. 3, No., 14, O the costructo of symmetrc oegatve matrx wth prescrbed Rtz values A. M. Nazar a, E. Afshar b a Departmet of Mathematcs, Arak Uversty, P.O. Box , Ira. b Departmet of Mathematcs, Khome Brach, Islamc Azad Uversty, Khome, Ira. Receved 5 July 14; Revsed 6 September 14; Accepted 3 September 14. Abstract. I ths paper for a gve prescrbed Rtz values that satsfy the some specal codtos, we fd a symmetrc oegatve matrx, such that the gve set be ts Rtz values. c 14 IAUCTB. All rghts reserved. Keywords: Rtz values, Noegatve matrx. 1 AMS Subect Classfcato: 15A9; 15A Itroducto The Rtz values of a matrx are the egevalues of ts leadg prcpal submatrces of order m = 1,,...,. I the symmetrc case, the Rtz values are real ad terlacg,.e. the ordered values λ at level ad λ +1 at the ext level + 1 satsfy λ +1 1 λ 1 λ +1 λ... λ Kostat ad Wallach [1,], costruct the uque ut Hesseberg matrx H wth gve R B, where R B s the lst of Rtz values. Beresford Parlett ad Glbert Strag [3] solved the problem for symmetrc arrow ad trdagoal matrces. Correspodg author. E-mal address: A-azar@araku.ac.r A. M. Nazar. Prt ISSN: 5-1 c 14 IAUCTB. All rghts reserved. Ole ISSN:

2 6 A. M. Nazar et al. / J. Lear. Topologcal. Algebra I the secto of ths paper smlar Theorem.1 of [4], we prove a theorem whch ca be used to costructo a oegatve matrx for a gve set of Rtz values a systematc method. The algorthm of ths method s preseted secto 3.. Ma Result We wll use some stadard basc cocepts ad results about square oegatve matrces such as reducble, rreducble, Frobeus ormal form of a reducble matrx, rreducble compoet ad Frobeus Theorem about the spectral structure of a rreducble matrx as are descrbed [1]. Theorem.1 Let B be a rreducble symmetrc oegatve matrx ad let R B = {λ 1 1 ; λ 1, λ ;...; λ 1, λ,..., λ be the Rtz values of B that satsfy the terlacg codto 1 for = 1,,..., 1. If λ λ a s the Perro egevalue of B ad A = a λ s a oegatve symmetrc matrx wth Rtz values {µ 1 ; µ, µ 3 }, the there exst a oegatve B sa symmetrc matrx C = s T wth the followg Rtz values a δ M = {λ 1 1 ; λ 1, λ ;...; λ 1, λ,..., λ ; λ 1, λ },..., λ 1, µ, µ 3 }, where s s the ormalzed egevector correspodg to the Perro egevalue λ δ = µ 3 + µ λ. Proof. Let s be the ormalzed egevector assocated to the Perro egevalue of B. We fd the 1 matrx V 1 such that Y 1 = s, V 1 be utary ad BY 1 = λ s, BV 1. Therefore B 1 = Y 1 BY 1 = λ ss s BV 1 λ V1 s V 1 BV = 1 λ,. B where B = V1 BV 1 s a 1 1 matrx ad {λ 1 1,..., λ 1 } s set of egevalues of B. O the other had, by Schur decomposto theorem there exst the utary matrx V of order 1, such that V BV = T B, where T B s upper tragular matrx wth the egevalues of B o ts ma dagoal. If 1 Y =. V, ad

3 A. M. Nazar et al. / J. Lear. Topologcal. Algebra whereas V s a utary matrx, the Y s also utary. Thus Y B 1 Y = Y Y 1 BY 1 Y = Y 1 Y BY 1 Y = Y BY, f Y = Y 1 Y Y = Y 1 Y = s V 1 V = s T = Y = s T. f T = V 1 V Sce T s a utary matrx, we ca wrte Y Y = ss + T T = I, Y Y = s s s T 1 T s T =, T I 1 ad Y BY = s Bs s BT T Bs T = BT λ TB = T B. 3 Cosequetly, T B s a upper tragular matrx wth egevalues of B o ts ma dagoal. O the other had by Schur decomposto theorem there exst a utary matrx X, such that X AX = T A, where T A s a upper tragular matrx so that egevalues of A {µ, µ 3 } le o ts ma dagoal. Assume that the matrces X ad X have the followg parttos: X = K1 K, X = K 1 K where K 1 ad K are 1 vectors. Sce X s a utary matrx we have, XX K1 K = 1 K 1K 1 K K1 K K =, 1 XX = K1 K + K K = I. 4 By 4 we have T A = X AX = K 1 λk 1 + K ak 1 + K 1 ak + λ K K = λk 1 K 1 + ak K 1 + K 1 K + λ K K. 5 Now we cosder two matrces Z ad Z ad a oegatve matrx C of order the followg forms: Z = sk T, Z K 1 K = s K1 T, C = B sa as. δ

4 64 A. M. Nazar et al. / J. Lear. Topologcal. Algebra Usg relatos ad 4, t s easy to show that Z s a utary matrx. Now by relatos 5, we ca calculate Z CZ as follows: Z CZ = K 1 λk 1 + ak K 1 + K1 K + λ K K 1 K 1 as T + K s BT T sak 1 + T BsK T = BT TA ˆT = T C, B where T C s a upper tragular matrx ad the elemets of ts ma dagoal are the elemets of the last level of M. O the other had by the above relato, C ad T C are smlar, therefore C solves the problem whch completes the proof. 3. A Algorthm for costructo of a symmetrc oegatve matrx Let R A = {λ 1 1 ; λ 1, λ ;...; λ 1, λ,..., λ }, where λ 1, λ,..., λ are the egevalues of leadg prcpal submatrx A for = 1,,, ad assume that R A satsfes the terlacg codto 1. Assume the egevalues of A +1 are oly dfferet wth egevalues of A λ +1, λ where λ s the Perro egevalue of A +1 ad λ = λ +1, for = 1,,..., 1 ad = 1,,...,. By followg algorthm we costruct a oegatve symmetrc matrx A wth Rtz values R A ad leadg prcpal submatrces. step 1 : Let A 1 = λ 1 1. Step : Let A = λ λ 1 1 λ 1 1 λ1 1 λ λ λ 1 1 λ1 1 λ 1 λ + λ 1 λ 1 1 By terlacg codto 1 the elemets of A are oegatve. Step : We costruct a oegatve symmetrc matrx wth prescrbed Rtz values {λ 1 1 ; λ 1, λ ;...; λ 1, λ,..., λ that satsfy terlacg codto 1 ad λ 1 1,..., λ 1 les the set of egevalues of A. At frst we costruct the symmetrc oegatve matrx A wth Rtz values {µ 1 ; µ, µ 3 }, where µ 3 = λ, µ = λ 1, } µ 1 = µ + µ 3 λ 1 1 µ1 ad A a = a λ 1 ad a = µ 3 µ 1 µ 1 µ. The codto o µ 1 s the solvablty codto for ths problem. Now by Theorem.1 we costruct A by 1 combg 1.

5 A. M. Nazar et al. / J. Lear. Topologcal. Algebra two matrces A 1 ad A the form A = correspodg to the Perro egevector of A. A 1 as as T µ 1, where s s the egevector Example 3.1 We costruct a matrx wth the followg prescrbed Rtz values {1; 1, 3; 1,, 4; 1,, 3, 5; 1,,, 4, 6}. A 1 = 1, A = λ 1 1 a a δ 1, where a = λ 1 1 λ 1 λ λ 1 1 =, δ 1 = λ + λ 1 λ 1 1 = 1. The costructo of the matrx A 3 eeds the ormalzed egevector to assocated egevalue 3. Ths vector s s = 1 1. We must costruct matrx A wth Rtz values {µ 1 ; µ, µ 3 } = {µ 1 ;, 4}, so that λ = 3 s les o the ma dagoal of A. µ1 a A = a λ, where µ 1 = µ + µ 3 λ = 3, a = µ 1 µ µ 3 µ 1 = 1. Now by Theorem.1, we combe A ad A to costruct A 3 as 1 / A 3 = 1 /. / / 3 Smlarly we ca costruct the matrces A 4 ad A 5. 1 / 1/ A 4 = 1 / 1/ / / 3 1/ 1/ 1/ 1/, 4 1 / 1/ /4 1 / 1/ /4 A 5 = / / 3 1/ 1/ 1/ 1/ 1/. 4 / /4 /4 1/ / 5 Refereces [1] Bertram Kostat, Nola Wallach, GelfadZetl theory from the perspectve of classcal mechacs I, Prog. Math [] Bertram Kostat, Nola Wallach, GelfadZetl theory from the perspectve of classcal mechacs II, Prog. Math [3] Beresford Parlett, Glbert Strag, Matrces wth prescrbed Rtz values, Lear Algebra ad ts Applcatos

6 66 A. M. Nazar et al. / J. Lear. Topologcal. Algebra [4] A. M. Nazar, F. Sherafat, O the verse egevalue problem for oegatve matrces of order two to fve, Lear Algebra Appl