Eigenvectors and Eigenvalues


 Alvin Wilkinson
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1 MTB ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions. The ses onsiere here re ivrite so they n e esily lulte n visulize s grphs. However, lultions re typilly extene into multivrite hyperspe, the ommon onition in multivrite sttistis. In ll ses, there re s mny eigenvet/eigenvlue pirs s there re imensions in the multivrite t so long s the t re of "full rnk" (i.e., not reunnt euse some vriles re liner omintions of the others). F extene isussion, see RA Johnson & DW Wihern Applie Multivrite Sttistil Anlysis 4th Eition 1998, AC. Renher Methos of Multivrite Anlysis Consier liner trnsfmtion from r to R: x r x r y r y r < R Mr F exmple, given liner trnsfmtion speifie y some mtrix M n t in vet X: M 5 3 r 5 3 Then: R Mr R 6 In generl there re two geometri interprettions of this trnsfmtion: Let M esrie the trnsfmtion suh tht R Mr: r R R r Using the left interprettion, suppose the (x,y) plne is overe y n elsti memrne whih my e shrunk, strehe, rotte so tht:  r (, ) eomes R (, ) f ny point on the plne Using the right interprettion, suppose the points o not hnge position, ut the referene system use to esrie their lotion is shifte y rigi rottion. In this se:  r (, ) n R (, ) re the sme ut the ointe in the referene system f r n R hve hnge
2 MTB 050 Using the left geometri interprettion ove, we my sk if there re ny vets r tht remin unhnge in iretion uring trnsfmtion? From Multivrite Wksheet MTB 040, we note tht in ft this is the se. These re terme the "prinipl" "hrteristi" iretions. Using the right interprettion ove, Eigenvets n e visulize s hving their foot t the enter of the referene system n he prllel to the iretions of referene system R. Note, however, tht we will llow the spe to e shrunk strethe in the Eigenvet iretions. The mount of this strething is lle the "prinipl" "hrteristi" vlues lulte f eh Eigenvet s n ssoite Eigenvlue. In lgeri terms, vet Eigenvet R r where Eigenvlue is some slr onstnt. R r R Mr r R To lulte eigenvlues n eigenvets, rememer tht: R Converting mtrix lger to simultneous equtions: x r x r y r y r xr y r 0 x r yr 0 Eigenvet eqution > 0 F & oth not equl to zero, the eigenvet eqution is solve using the eterminnt: ( 1) ( ) ( ) 0 < Chrteristi eqution of M < Qurti eqution of the generl fm: A B C 0
3 MTB Generl solution to the qurti: Thus: ( 1) ( ) hs two generl solutions: ( ) 0 B B 4A C A B B 4A C A ( ) ( ) 4( 1) ( ) ( 1) ( ) ( ) 4( 1) ( ) ( 1) To lulte eigenvets, reple eh solve eigenvlue k into the Eigenvet eqution. Note: Equtions will e reunnt n will therefe speify the eqution of line. As result, it is ustomry give eigenvets unit length. Things to rememer out eigenvets:  Eigenvets foun from M re not neessrily perpeniulr.  If M is symmetri [i.e., M M T ] then eigenvets re perpeniulr. Only in this se is the righthn piture ove is vile geometri interprettion. Exmple Clultion: Solving f Eigenvlues: M ( ) ( ) 4( 1) ( ) ( 1) ( ) ( ) 4( 1) ( ) 0.68 ( 1) Solving f Eigenvets: 1. E 0 ^ Eigenvet eqution Plugging Eigenvlues k into the Eigenvet eqution results in expressions tht give the x ointe of eh Eigenvet in terms of y vie vers. This is the eqution f line n implies onstnt rtio etween y n x f eh Eigenvet. F Eigenvet 1: F Eigenvet : 1. x y 0 x 0 x 1 y x y ^ rtios of x in terms of y ^
4 MTB Mth's uiltin funtions: evls eves eigenvls( M) eigenves( M) eigenve M evls 1 evls eves eigenve M evls < Vet of eigenvlues Mtrix of eigenvets, eh olumn < is n eigenvet ssoite with one of the eigenvlues ove. Generlly, eh olumn is in the sme er s the elements in the eigenvlue vet. < A wy to hek whih eigenvet elongs with whih eigenvlue! Note tht Eigenvet 1 n Eigenvet hve the sme rtios of x versus y vlues s lulte ove: F Eigenvet 1: F Eigenvet : ^rtio ^ y ^x ^rtio ^ y ^x Prototype in R: #EIGENVECTORS AND EIGENVALUES: #MAKING MATRIX M: Mmtrix((5,, 3,),nrow,nol,yrowT) > M [,1] [,] [1,] 5  [,] 3 > #COMPUTING EIGENVALUES AND EIGENVECTORS OF MATRIX M: Eeigen(M) > E $vlues [1] $vets [,1] [,] [1,] [,] Length of repte Eigenvets: F Eigenvet 1: F Eigenvet : E 1 eves E 1 E eves E T T L E1 E 1 E1 L E1 1 L E E E L E 1 Beuse solution of the Eigenvet eqution onlesults in eqution of lines, Eigenvets hve etermine iretion (rtio of x versus y) ut re ritrry in length. It is ustomry to sle Eigenvets to unit length. As you n see, oth MthC n R i this, resulting in the numeril vlues repte f the Eigenvets in eh se.
5 MTB Displying Eigenvets: Here f visul effet we use less extreme mtrix M from Multivrite Wksheet MTB 040: (funtion mke.gri() inlue in R sript) Xmke.gri(10,10) Mmtrix((1.0,0.3,0.3,1),nrow,nol,yrowT) Eeigen(M) M #LINEAR TRANSFORMATION: ZM%*%t(X) Zt(Z) #PLOT TRANSFORMATION VECTORS: plot(z,type'n',xl'x',yl'y') rrows(x[,1],x[,],z[,1],z[,],ol'purple',lw1,oe,length0.05) rrows(0,0,8*e$vets[,1],8*e$vets[,],ol're',oe,length0.05) Y X Eigenvets in re (sle to length of 8 f isply effet).
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