Iteratioal Joural of Mathematical Archive-5(7), 04, 49-56 Available olie through wwwmaifo ISSN 9 5046 APPROXIMAION OF SMALLES EIGENVALUE AND IS CORRESPONDING EIGENVECOR BY POWER MEHOD Alaur Hussai Laskar ad Samira Behera* Departmet of Mathematics, Assam Uiversity, Silchar- 7880, Idia (Received O: 6-06-4; Revised & Accepted O: 9-07-4) ABSRAC Power method is ormally used to determie the largest eigevalue (i magitude) ad the correspodig eigevector of the system AX = λx I this study, we eamie power method for computig the smallest eigevalue ad its correspodig eigevector of real square matrices Our work is based o choosig of iitial vector i power method for acceleratio purpose Fially, we illustrate the method with eample ad results discussed Keywords: Domiat eigevalue, power method, Adjoit of a square matri, Iverse matri i terms of Adjoit matri INRODUCION We study the problem of calculatig the eigevalues ad eigevectors If oly a few eigevalues are to be calculated, the the umerical method will be differet tha if all eigevalues are required Eigevalues ad eigevectors play a importat part i the applicatios of liear algebra he aive method of fidig the eigevalues of a matri ivolves fidig the roots of the characteristic polyomial of the matri I idustrial sized matrices, however, this method is ot feasible, ad the eigevalues must be obtaied by other meas Fortuately, there eist several other techiques for fidig eigevalues ad eigevectors of a matri, some of which fall uder the realm of iterative methods hese methods work by repeatedly refiig approimatios to the eigevectors or eigevalues, ad ca be termiated wheever the approimatios reach a suitable degree of accuracy Iterative methods form the basis of much of moder day eigevalue computatio he geeral problem of fidig all eigevalues ad eigevectors of a o-symmetric matri A ca be quite ustable with respect to perturbatios i the coefficiets of A, ad this makes more difficult the desig of geeral methods ad computer programs he eigevalues of a symmetric matri A are quite stable with respect to perturbatios i A he eigevalues of a matri are usually calculated first, ad they are used i calculatig the eigevectors, if these are desired he mai eceptio to this rule is the power method described i this paper, which is useful i calculatig a sigle domiat eigevalue of a matri For obtaiig eigevalues ad eigevectors for low order matrices, ad 3 3 his ivolved firstly solvig the characteristic equatio det( A λi) = 0 for a give matri A his is a th order polyomial equatio ad, eve for as low as 3, solvig it is ot always straightforward For large eve obtaiig the characteristic equatio may be differet Cosequetly, i this paper we give a brief itroductio to alterative method, essetially umerical i ature, of obtaiig eigevalues ad perhaps eigevectors Algebraic procedures for determiig eigevalues ad eigevectors are impractical for most matrices of large order Istead, umerical methods that are efficiet ad stable whe programmed o high-speed computers have bee developed for this purpose Such methods are iterative, ad, i the ideal case, coverge to the eigevalues ad eigevectors of iterest I this paper, we outlie power method, ad summarize derivatios, procedures ad advatages he method to be eamied is the power method I sectio of this paper, we have discussed some basic cocepts regardig eigevalues ad eigevectors with eample required to uderstad the cocepts that are discussed I sectio 3, we have preseted power method with eample for approimatig smallest eigevalue ad its correspodig eigevector of the real square matri A Fially, i sectio 4, we summarized some cocludig remarks that are used i practice Correspodig author: Samira Behera* Departmet of Mathematics, Assam Uiversity, Silchar- 7880, Idia E-mail: samirabehera998@gmailcom Iteratioal Joural of Mathematical Archive- 5(7), July 04 49
For the purposes of this paper, we restrict our attetio to real-valued, square matrices with a full set of real eigevalues PRELIMINARIES I this sectio, we recall some basic cocepts which would be used i the sequel Defiitio : he mior of a elemet of a determiat of order greater tha oe is the determiat of et lower order obtaied by deletig the row ad the colum of the give determiat i which the elemet occurs he mior of the elemet a i the determiat A is deoted by M Defiitio : he cofactor of a elemet of a determiat of order greater tha oe is the coefficiet of that elemet i the epasio of the determiat he cofactor of the elemet a i A is deoted by A he cofactor of a elemet a i A ca be determied i terms of its mior as A = ( ) i+ j M Defiitio 3: Correspodig to a square matri cofactor of a i A he which is deoted by AdjA A = a, we form a matri 04, IJMA All Rights Reserved 50 B= A, where A is the B (traspose of B ) is called the Adjoit Matri or Adjugate Matri of the Matri A Defiitio 4: A square matri A is ivertible if ad oly if A is o-sigular Defiitio 5: Let λ, λ,, λ be the eigevalues of a matri A λ is called the domiat eigevalue of A if λ > λ i, i =,, he eigevectors correspodig to λ are called domiat eigevectors of A Defiitio 6: Eigevectors correspodig to distict eigevalues are liearly idepedet However, two or more liearly idepedet eigevectors may correspod to the same eigevalue () () (3) A, A, A, be a sequece of matrices i m Defiitio 7: Let R We say that the sequece of matrices m coverges to a matri A R if the sequece A ( k ) i, j of real umbers coverges to A i, j for every pair i m, j, as k approaches ifiity hat is, a sequece of matrices coverges if the sequeces give by each etry of the matri all coverge 8 EIGENVALUES AND EIGENVECORS Cosider the equatio AX = λ X () Here, A is a matri, λ is a scalar ad X is a o-zero vector he solutio of () requires the solutio of λ he scalar λ (real or comple) is called the eigevalue or latet root or characteristic value of A X is called the correspodig eigevector or characteristic vector of the matri A he eigevalues of a matri are of great importace i physical problems hey occur i the aalysis of stability ad i the equatios of vibratios i structures or electrical circuits he stability of a aircraft is determied by the locatio of the eigevalues of a certai matri i the comple plae If A= a, the () ca be writte as a λ a a3 a a a λ a3 a X = 0 a a a3 a λ ()
Or[ A λi] X = 0 (3) his is a set of liear homogeeous equatios It will have a o-trivial solutio if ad oly if A λi = 0; that is, if ad oly if a λ a a3 a a a λ a3 a = 0 a a a a λ 3 he determiat is a polyomial of degree i λ he polyomial is called the characteristic polyomial of matri A It is usually deoted by P( λ ) he roots of this polyomial are the eigevalues (or the latet roots or the characteristic values) of the matri A If the values of λ are λ, λ,, λ, which may ot all be distict, the the eigevectors of the matri A are give by AX = λx, AX = λx,, AX = λx (5) Hece the determiatio of eigevalues of a matri A is othig but solvig a algebraic equatio of degree 9 WO IMPORAN PROPERIES OF EIGENVALUES AND EIGENVECORS Property : If X is a eigevector of A correspodig to the eigevalue λ ad A is ivertible, the X is a eigevector of correspodig to its eigevalue λ Property : If A is a o-sigular matri, the eigevalues of are the reciprocals of the eigevalues of A Proof: Let λ be a eigevalue of A ad X be a correspodig eigevector he AX = λ X X = A ( λx) = λ( A X) X = A X λ ( A is o-sigular λ 0) A X = X λ is a eigevalue of A ad X is a correspodig eigevector λ Coversely, suppose that k is a eigevalue of Sice A is o-sigular is o-sigular ad ( A ) = A, therefore it follows from the first part of this property that k is a eigevalue of A hus each eigevalue of is equal to the reciprocal of some eigevalue of A Hece the eigevalues of the reciprocals of the eigevalues of A Eample : Let us ow cosider the matri A = 5 04, IJMA All Rights Reserved 5 (4) are othig but to fid the eigevalues ad the correspodig eigevectors by direct method ie by algebraic procedures for verifyig the above two properties Solutio: he characteristic equatio is P( λ) = A λi = 0 λ = 0 5 λ
λ = ad λ = Which gives he correspodig eigevectors are obtaied thus: (i) For λ = Let the eigevector be X = he we have A = = 5 = ad + 5 = which gives the equatios = 4 which gives Hece the eigevector for eigevector is X = [ 4,] (ii) For λ = Let the eigevector be X = A = = 5 λ = is X [ 4, ] = Sice is arbitrary, we ca take = ad hece the he we have = ad + 5 =, which gives = 3 which gives the equatios Hece the eigevector for λ = is X [ 3, ] eigevector is X = [ 3,] hus, the eigevalues are 4, X = ad X = 3, respectively [ ] [ ] = Sice is arbitrary, we ca take = ad hece the λ = ad λ = ad the correspodig eigevectors are Now, to fid eigevalues ad eigevectors of the matri, we eed to fid ad for that we proceed as follows: Clearly, the matri A is o-sigular AdjA 5 = AdjA 5 A = = ( A ) A = 5 6 = 04, IJMA All Rights Reserved 5
he characteristic equatio of the matri P( λ) = λi = 0 5 λ 6 = 0 λ is which gives λ = ad λ = he correspodig eigevectors are obtaied thus: (i) For λ = Let the eigevector be X = he we have A = 5 6 = which gives the equatios 5 + 6 = ad =, which gives = 4 Hece the eigevector for λ = is X = [ 4, ] Sice is arbitrary, we ca take = ad hece the eigevector is X = [ 4,] (ii) For λ = Let the eigevector be X = = A 5 6 = he we have 5 + 6 = ad =, which gives = 3 which gives the equatios Hece the eigevector for λ = is X [ 3, ] eigevector is X = [ 3,] = Sice is arbitrary, we ca take = ad hece the From the above eample we have see that if X is a eigevector of A correspodig to the eigevalue λ ad A is ivertible, the X is a eigevector of eigevalues of correspodig to its eigevalue λ Also, we have see that the are the reciprocals of the eigevalues of A (obviously, the matri A is o-sigular ie A 0 04, IJMA All Rights Reserved 53
3 HE POWER MEHOD (IERAIVE MEHOD) his method is used for eigevalue problems where very few roots of the characteristic equatio are to be foud Let all the eigevalues be distict A arbitrary vector Y ca be epressed as Y = ax + ax + + ax = ax r r= r () o fid the umerically largest or domiat eigevalue ad its associate eigevector, we start with a arbitrary vector,0,0 It Y he vector is multiplied successively by the matri A A coveiet choice for ca also be take as [ 0,], [,], [ ] procedures require a iitial estimate of the quatity sought to be take Multiplyig the equatio () by A, we get () Y = AY = arax r = arλrx r r= r= () () Multiplyig by A agai ad lettig Y Y is [, 0] or [ ], or ay other vector of the correct size It must be oted that all iterative = AY, we get () = rλr r = rλr r = rλr r r= r= r= Y A a X a AX a X Proceedig like this, we get at the mth iteratio m Y = arλr Xr r= m m m = aλ X+ aλ X + + aλ X Suppose λ is the largest eigevalue he, m m m Y = λ [ ax + a( λ λ ) X + + a( λ λ ) X ] m he values ( i ) ( i ) m herefore, Y λ ax, a scalar multiple of λ λ ted to zero as m ad hece all the terms become egligible ecept the first term ( m ) m X, as m Also, Y + aλ + X for large m ( m+ ) Y ( m+ ) Y ad Y, we get λ for large m, the required Y herefore, takig the ratio of the magitudes of largest eigevalue It is clear that the rate of covergece depeds o the ratio of the moduli of the two largest eigevalues Whe this ratio is early uity, the covergece is very poor o avoid this, the followig procedure is adopted: (i) he arbitrary vector Y is selected such that the largest elemet of this vector is uity; ie the vector Y is put ito the ormalized form with the largest elemet uity (ii) he ormalized vector is multiplied by A (iii) he ew vector is ormalized by dividig each elemet by the largest elemet Let this largest elemet be l m (iv) he process is repeated util the values of l m ad l m + differ by some prescribed small value he value of l m gives the value of the largest eigevalue ad the vector Y is the eigevector correspodig to l m 3 SMALLES EIGENVALUE AND IS CORRESPONDING EIGENVECOR BY POWER MEHOD We have already stated that the eigevalues of, if A is o-sigular, are the reciprocals of the eigevalues of A herefore, the smallest eigevalue of A is the largest eigevalue of Hece we ca use the power method to determie the smallest eigevalue of A by workig with istead of A his procedure is illustrated i eample 04, IJMA All Rights Reserved 54
Eample : Let us ow cosider the same matri of eample ie eigevalue ad its correspodig eigevector by applyig power method to Solutio: Here, A = 5 istead of A 5 6 A = 5 We kow that from eample, = Let us fid the largest eigevalue of 35 7 Z = A ξ0 =, 05, 05 α = ξ = 5 46 Z = A ξ =, α 5, ξ 5 = = 0 55 430 Z3 = A ξ =, α3 3, ξ3 3 = = 000 4575 403 Z4 = A ξ3 =, α 5 4 = 5, ξ4 = 000 457 4050 Z5 A ξ 4, α5 05, ξ = = = 5 = 05 000 45 404 Z6 A ξ 5, α6 05, ξ = = = 6 = 05 000 4060 40 Z7 = A ξ6 =, 7 0, 7 0 α = ξ = 000 407 4006 Z8 = A ξ7 =, α8 = 005, ξ8 = 005 000 to approimate the smallest by power method We begi with a iitial approimatio [ ] ξ 0 =, All these computatios show that α, α, coverges to, which is the largest eigevalue of ξ ξ ξ coverges to [ ] 0,,, ad X = 4, is the correspodig eigevector Sice the eigevalues of A are the, the smallest eigevalue of A is his is the same as the result we obtaied earlier (i reciprocals to those of eample by direct method ie by algebraic procedures) We have got the correspodig eigevector also the same as the oe obtaied earlier (i eample by direct method ie by algebraic procedures) 4 CONCLUSION I this paper, we have studied power method to approimate the smallest eigevalue ad its correspodig eigevector of real-valued square matrices Here, we used the ew iitial vector for the power method Maily, i this paper we have see that with eamples ad, if we apply the power method to, we will get the approimate largest eigevalue of ad its correspodig eigevector ad cosequetly we will get the approimate smallest eigevalue of A with the same eigevector as if X is a eigevector of A correspodig to the eigevalue λ ad A is ivertible, the X is a eigevector of correspodig to its eigevalue his approimate smallest λ eigevalue ad its correspodig eigevector appear to be approachig the eact smallest eigevalue ad its correspodig eigevector as we have obtaied earlier i eample by direct method ie by algebraic procedures 04, IJMA All Rights Reserved 55
REFERENCES [] GHGolub, CF Va Loa, Matri Computatios, Johs Hopkis Uiversity Press, 996 [] Richard Broso, Schaum s outlie of theory ad problems of matri operatios, McGRAW-HILL BOOK COMPANY [3] Devi Prasad, A Itroductio to Numerical Aalysis, hird Editio:006, Narosa Publishig House Pvt Ltd [4] C Moler, G Stewart, a Algorithm for Geeralized Matri Eigevalue Problems, SIAM J Numer Aal,Vol 0, No, 973 [5] Kedall E Atkiso, A Itroductio to Numerical Aalysis, Secod Editio: 988, Joh Wiley ad Sos Publishers [6] MK Jai, SRK Iyegar, RK Jai, Numerical Methods for Scietific ad Egieerig Computatio, Fourth Editio: 003, New Age Iteratioal (P) Limited, Publishers [7] GW Stewart, Itroductio to Matri Computatios, Academic Press, New York, 973 [8] GH Golub, CF Va Loa, Matri Computatios, Johs Hopkis Uiversity Press, Baltimore, 996 [9] JB Fraleigh, AR Beauregard, Liear Algebra, Addiso-Wesley Publishig Compay, 995 Source of support: Nil, Coflict of iterest: Noe Declared [Copy right 04 his is a Ope Access article distributed uder the terms of the Iteratioal Joural of Mathematical Archive (IJMA), which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited] 04, IJMA All Rights Reserved 56