 Applied Mathematics, 2011, 2, 1443-1445 doi:10.4236/am.2011.212204 Published Online December 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Inverse Eigenvalue Problem for Gen eralized Arrow-Like Matrices Zhibin Li, Cong Bu, Hui Wang College of Mathematics, Dalian Jiaotong University, Dalian, China E-mail: lizhibinky@163.com Received October 14, 2011; revised November 15, 2011; accepted November 23, 2011 Abstract This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two character-istic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided. Keywords: Generalized Arrow-Like Matrices, Characteristic Value, Inverse Problem, Unique 1. Introduction The Inverse eigenvalue problem for matrices in the pro- blems involved in the field of structural design, pattern recognition, parameter recognition, automatic control and so on, it has a good engineering background, and its research has obvious significance . Many experts and scholars have addressed more extensively and in-depth studied, get a lot of conclusions about inverse eigenvalue problem for Jacobi matrices , but there is less research about the inverse eigenvalue problem for arrow- like matrices [3,4]. This paper researches the following in-verse eigenvalue problem for generalized arrow-like ma-trices. Generalized arrow-like matrices refer to the matrix as follows: 11 1121111211mmmmmmmmmnnnabb bcacaJcabcabca (1) When , 1mJ becomes generalized Jacobi matrix ; when , mnJ is an arrow-like matrice. This article studies the following characteristic value inverse: Question IEPGAM. Given two real numbers ,( )  and two nonzero real vectors T12(, ,, ),nnxxx xRT12(, , ,)nnyyy yR. Find the nn real generalized arrow-like matrice J, ch that su,JxxJyy(,). The expression and an algorithm of the solution of th e problem is given in Section 2, and a numerical example is provided in Section 3. 2. The Solution of Question IEPGAM Because x and (,)y are two characteristic pairs of the generalized arrow-like matrices J, In it, Let: 010nnxx yy100 0nnbbcc , (2) 11(0,1,,)iiiiixyDixyn, (3) 11(2,3,, 1iiixxEimyy ) (4) So 11 12111+b mm mmaxxbxb xx, (5-1) 112 22+cx axx, (5-2) … 11+mmmcxax xm111 12mmm mmaxb x, （5-m）1+mcx x2 22 3m mm max bx, (5-m+1) 11+mmcx x2m , (5-m+2) … Z. B. LI ET AL. 1444 122 111+n nnnnnncx axbxx 11+nn nncx ax, (5-n–1) nx . (5-n) 11 12111+b mm mmayybyb yy 112 22+cy ayy, (6-1) , (6-2) … 11mmmcyay ym, (6-m) 11112mmmmm mcyaybyy1 112 22 3mmm mm mmcy aybyy, (6-m+1) 2  , (6-m+2) … 2211 1nnnn nnncyay byy111nn nnncy ayy, (6-n–1) . (6-n)  For inverse . ,(1,2,,1),iibci mmn ,3,,)m n(2ai miFrom (5) and (6), we can get 11 1+(2,iiii iiicxaxbxxi mmn3,,) 11 1+(2,iiiiiiicyay byyimmn, (7) 3,,) . (8) In order to eliminate , multiply by i on both sides of (7), multiply by iia yx on both sides of (8), then cut on both sides, we can get i11=()+(2,3,, )iiiiibDxyc Dimmn. (9) To problem A, because ,(2,3, ,1)iickbin, so (9) become 1()(2,3,,)iiiii ibDxykb Dim mn  1 (10) Let , because , so in0nDn11 ()nnnxybD k1in, Let , -1 122 2()nnnnnnxyx ybD kk ; ………… Let , 2im11+2 211 (1) (2)()nnn nmmmm nmnmxyx yxybD kkk    Under normal circums ta n ce s, (1)()0()( 1,2,,nj ns nsjj nsjsxybDj mmnk  1). (11) If , then 0 (1,2,,1)jDjmm n ,iixy can not be zero at the same time ,so (1)()0() (1,2,,nj ns nsjnsjsjxybjmmDk 1)n, (12) c=,(1,2, ,1)jjkbj mmn , (13) 11 111 1,0,0(2,3,,)jjjjj jjjjjjjj jjxcxbx xxaycybyyyjm mn   ;. (14)  For inverse ,,. 1mmmFrom (5) and the m + 1 equation o f (6), acb+11 111=( )+mmm mmmcExyb D , (15) 1m+111111m+2=mmmmaE xyxybE , (16) mmcbk. (17)  For inverse 1,,(2,3, ,1)iiccbi m,,)m, (2,3aii. From (5) and 2 to m equation of (6), 11+(2,3,iiiicxax xim,), (18) 11+(2,3,iiiicyay yim,). (19) From (18) and (19), we can get 1i=()(2,3,,)iiicE xyim, (20) i11=(2,3,iiiaExyxy im,), (21) (2,3,, 1)iicbi mk. (22)  For inverse ,ab. 11From (5) and (6), we can get 11 12112+msssax bxxbx, (23) 11 12112+msssay byyby. (24) If 10D, from (23) and (24), then we can get 1221122 1211()mss ssxyxyb xyxyaD , (25) 11111 1211mssssxyb yxyxbD . (26) According to the above analysis, to question IEPGAM, we can get the follow theorem. Theorem. If the following conditions are satisfied: 1) 10D; 2) 0 (Di1,2, ,1)imm n 0 (2, 3,,1)Ei m; 3) i Then question IEPGAM has the unique solution, and Copyright © 2011 SciRes. AM Z. B. LI ET AL. Copyright © 2011 SciRes. AM 1445(1)()0() (1,2,,nj nsnsjnsjsjxybjmmDk 1)n(27) 112 311311() ()74xyb yxyxbD  ; 11 1111,0,0(2,3,,)jjjjjjjjjjjjjjjxcxbx xxaycyby yyjm mn   2232ckb , ;, (28) 3313ckb , 440ckb, 2212()=0xycE; 111 1+1()+=mmm mmmxy bDbkE , (29) 12 231172bx bxax, 1111 1m+21+1=mm mmmxyxyb EaE , (30) 12 2122=1xy xyaE, 11+1()=(2,3,jjjjxybikE,1)m, (31) 13313 4338=3xyxy bEaE, 111111211() (mssssxyb yxyxbD) ( 32) 43345444=3xcxbxax, 11=(2,3,jjjjxy xyajE,)m, (33) 5445655=1xcxbxax. 11111111,0,0msssmsssjbxxxabyyySo 7730024400010381002361400 033000 01J ;) (34) =,(2,3,,1jjckbin, (35) 2212()=xycE. (36) and ,JxxJyy. 3. Numerical Examples 4. References Example 1. Give 1,2, 2,2, 5,kmn TT(1,0,2,1,0) .  D. J. Wang, “Inverse Eigenvalue Problem in Structural Dynamics,” Journal of Vibration and Shock, No. 2, 1988, pp. 31-43. (1,1,1,1,1) ,xy It is easy to be calculated 12410,30, 10DDD  ;  H. Dai, “Inverse Eigenvalue Problem for Jacobi Matri-ces,” Computation Physics, Vol. 11, No. 4, 1994, pp. 451-456. 23410, 10,2.EEE From Theorem, the question IEPGAM has the unique solution. And 55 443231+=6xy xybDkk,  C. H. Wu and L. Z. Lu, “Inverse Eigenvalue Problem for a Kind of Special Matrix,” Journal of Xiamen University (Natural Science), No. 1, 2009, pp. 22-26. 2333331() 4bxybDkE3,  Q. X. Yin, “Generalized Inverse Eigenvalue Problem for Arrow-Like Matrices,” Journal of Nan Jing University of Aeronautics & Astronautics, Vol. 34, No. 2, 2002, pp. 190-192. 55440xybDk,