Where do eigenvalues/eigenvectors/eigenfunctions come from, and why are they important anyway?

Transcription

1 Where do eigevalues/eigevectors/eigeuctios come rom, ad why are they importat ayway? I. Bacgroud (rom Ordiary Dieretial Equatios} Cosider the simplest example o a harmoic oscillator (thi o a vibratig strig) d ux ux ( ) + ( ) 0 dx We ow this equatio has the liearly idepedet solutios u ( x) si( x) u ( x) cos( x) where is related to the requecy o oscillatio. As icreases, the requecy o oscillatio icreases. We ca gai more isight by imposig boudary coditios u(0) u( L) 0. he homogeeous solutios tae the orm o ux ( ) Au( x) + Bu( x) Asi( x) + Bcos( x) Applyig the irst boudary coditio, u (0) 0, we see that 0 u(0) B Usig the secod boudary coditio, we coclude that

2 0 ul ( ) Asi( L) Which meas that L π or iteger values o,,,3,... his gives us a iiite sequece o simple discrete values π L Writig the simple harmoic oscillator equatio as we see that it is o the orm d D [ u] u( x) u( x) dx Lu βu where β is a costat, a special value, called a eigevalue. (he word eige meas characteristic, or sigular, or uique i Germa.) II. Liear Algebra Applicatio I R, the ier product o two colum vectors is give by v v uv. u v ( u, u,..., u). uv + uv uv... v For symmetric matrices, A A, we ca say somethig more the vectors (eigevectors) associated with distict eigevalues are orthogoal. Au u Au u Multiplyig the irst equatio by ad the secod by we get u u u Au u u u. u u Au u u u. u Sice all o the expressios are scalars, we have u Au ( u Au) u A u Subtractig the two equatios we have the idetity

3 u ( A A ) u ( ) u. u I A is symmetric, that is i A leads to the ollowig results A, ad i the we must have uu. 0. his heorem: I A is symmetric ( A A ) the eigevectors correspodig to dieret eigevalues are orthogoal. Corollary: I a x symmetric matrix has distict eigevalues, the it has liearly idepedet (ad orthogoal) eigevectors. his set o eigevectors orms a basis. III. Bases o eigevectors I a matrix A has a complete set o eigevectors that ca be used as a basis, the solvig a liear system Au becomes very simple. { Let e, e,..., e} be a basis o eigevectors, which have bee ormalized with legth. Give a right had side R, we ca expad i terms o coordiates e he coordiates,, are easy to compute, sice by orthogoality e. e. e j j j Similarly, the solutio has a basis expasio u u e where the basis coeiciets are ot ow. Substitutig these ito Au, we get e Au u Ae u e Sice the basis coeiciets (coordiates) are uique, we must have u u,

4 So, the solutio o the liear system amouts to divisio! e u ue e IV. Geometric Iterpretatio o Eigevalues I you loo at the image o the uit ball, u, uder the mappig A, the the largest eigevalue is equal to the maximum stretchig o A. max max u he eigevector is the directio o maximum stretchig. V. Calculatio o Eigevalues ad Eigevectors or a iite dimesioal matrix From the deiitio o a eigevalue ad a eigevector, we have Ax x Ix which leads to 0 Ax Ix ( A I) x. his ca have o-zero solutios oly i the matrix is sigular (ot ivertible). hereore, we must have det( A I) 0. his results i a polyomial equatio o order or a x matrix. VI. Diagoalizatio o Liear Operators Aother way to view the behavior o eigevalues is the process o diagoalizatio. I we d cosider the dieretiatio operator, D, we see that dx ix ix D e ie ix So the uctio e is a eigeuctio, with eigevalue i. he eigeuctios are orthogoal with respect to the ier product Au π π imx ix imx ix i( m ) x e, e e e dx e dx π π 0 0 x π ( m x i m x x 0 ), m cos(( ) ) + si(( ) ) π 0, m We ca calculate the coordiates (i.e. Fourier coeiciets) by meas o the ormula π ix ix u u( x), e u( x) e d π x 0

5 With respect to this basis, the dieretial operator D is diagoal e ie 0 i ie e ie 0 0 i ie D e ie i... ie ix ix ix ix ix ix ix ix ix Sice the upper let coeiciet is 0, the matrix is sigular. We ow that D is ot -, sice D[]0 ad 0. VII. Filterig I you loo at the Fourier coeiciets o a cotiuous uctio, they geerally will loo lie the ollowig u he coeiciets will gradually decay. Large values o are associated with higher requecy compoets, usually with oise. Elimiatig these compoets ( low pass ilterig ) geerally leads to a smoother uctio. VIII. Eigeuctios o a Liear Operator I we have a liear operator, L, with a complete set o orthoormal eigeuctios, ad a ihomogeeous equatio

6 Lu we ca solve this problem usig a basis o eigeuctios. We ca expad i terms o the basis ( x) φ ( x) (with ow coeiciets, give by < φ, > ) ad the solutio give asu( x) uφ ( x). Substitutig this ito the ihomogeeous equatio, we get φ φ ( x) ( x) Lu L( u φ ( x)) u ( x) Equatig the coeiciets o φ ( x), we get the relatio u Sice, ad are bouded (i act they go to zero also), we must have u 0 he more quicly the coeiciets go to zero, the smoother (more dieretiable) the uctio. IX. Summary I a x matrix is symmetric ad has distict eigevalues, the it has complete set o eigevectors, which may be used as a orthoormal basis. Eigevectors ad eigeuctios are most ote idetiied with udametal modes o vibratio or oscillatio. Eigevalues are associated with the requecies o vibratio or oscillatio. he amplitude o the oscillatio goes dow asymptotically as the requecy icreases. he aster the rate o the decay, the smoother the uctio. Noise is geerally associated with high requecy compoets. Cuttig o high requecy compoets (settig the Fourier coeiciets to zero) results i a smoother uctio.