Liear lgebra Lecure #9 Noes This week s lecure focuses o wha migh be called he srucural aalysis of liear rasformaios Wha are he irisic properies of a liear rasformaio? re here ay fixed direcios? The discussio ceers o he eigevalues ad eigevecors associaed wih a marix he defiiios, calculaios, ad applicaios Ivaria direcios, eigevecors, ad eigevalues Le be a marix represeig a liear rasformaio T : R R re here ay ivaria direcios for his liear rasformaio? Tha is, ca we fid a vecor v such ha T ( v ) is parallel o v? This is a example of a irisic propery of he rasformaio somehig ha exiss idepede of wha basis is used or he coordiaes relaive o ha basis For example, a roaio i R has a axis of roaio regardless wha basis is used o describe he roaio For a orhogoal projecio oo some subspace V R, vecors i V remai uchaged, ad vecors i is orhogoal compleme are se o he zero vecor gai, his has ohig o do wih wha basis is used o represe his liear rasformaio The quesio of wheher we fid a vecor v such ha T ( v ) is parallel o v ca be rephrased as wheher here s a vecor v such ha T ( v) v v for some scalar This leads o he followig defiiio: Defiiio: If is a marix, we call a vecor v a eigevecor of if T ( v) v v for some scalar This scalar is called he eigevalue associaed wih he eigevecor Fidig he eigevalues ad eigevecors We ca rewrie v v as v Iv which is more coducive o usig algebra We ca he wrie his as Iv v or ( I v ) I order for a vecor v o be a eigevecor, i mus be i he kerel of I for some appropriae choice of This ca oly happe if his kerel is orivial which meas ha he marix I would have o o be iverible, ad we kow from our discussio of deermias ha a ecessary ad sufficie codiio for a marix o o be iverible is ha is deermia mus be equal o Tha is: v is a eigevecor of ( I ) v de( I ) s we ll see, if is a marix p( ) de( I ) will be a h degree polyomial i called he characerisic polyomial of So I will have a orivial kerel if ad oly if is a roo of his characerisic polyomial The eigevalues are herefore he roos of he characerisic polyomial Defiiio: The se of all eigevalues of a marix is called he specrum of Sice he eigevalues are he roos of a h degree polyomial, he specrum will cosis of a mos values These may be real umbers or complex umbers, possibly wih repeiio, ad ay complex eigevalues mus occur i complex cojugae pairs [This follows from he Fudameal Theorem of lgebra ay polyomial wih real coefficies ca, i heory, always be facored io a produc of liear facors a irreducible quadraic facors, ad hese irreducible quadraic facors will yield complex cojugae pairs (by he quadraic formula)] If a eigevalue occurs as a repeaed roo of he characerisic polyomial, we refer o he mulipliciy of he roo as he algebraic mulipliciy of he eigevalue Defiiio: If is a eigevalue of, he ker( I ) will be a subspace called he eigespace of, or E s wih ay subspace i mus be closed uder scalig ad vecor addiio This yields he followig wo corollaries: Corollary : If v is a eigevecor associaed wih a eigevalue, he v will also be a eigevecor for ay scalar Corollary : If v ad v are eigevecors associaed wih he same eigevalue, he cv+ cv will also be a eigevecor for ay scalars c, c revised /6/7
Defiiio: The geomeric mulipliciy of a eigevalue of a marix is dim[ker( I )], ie he umber of liearly idepede eigevecors associaed wih his eigevalue Example: Fid he eigevalues ad eigevecors of he marix Soluio: We calculae I, so he characerisic polyomial is p ( ) de( I ) de ( ) 6+ 8 This is easily facored o give p ( ) ( 4)( ), so he eigevalues are 4 ad How you order hese does maer, bu you should keep he idexig cosise For each eigevalue, we ex is eigevecors, ie ker( I ) for each eigevalue i : 4 4 gives I 4, so ker( I ) is foud by row reducio This gives x x or x, so if we le v, his spas he eigespace E 4 gives I, so ker( I ) is foud by row reducio This x gives x or x, so if we le v, his spas he eigespace E I he example, we had wo disic, real eigevalues which produced wo liearly idepede eigevecors which may be used as a basis for R, a eigebasis Wha is he marix of his liear rasformaio relaive o v 4v 4 he special basis? The relaios [ ] D, a diagoal marix If we wrie v v S v v, he S, ad [ ] S S D This will be he case for ay marix for which we ca produce a eire basis cosisig exclusively of eigevecors This moivaes he followig: Defiiio: marix is called diagoalizable if i is possible o fid a basis for eigevecors of If is diagoalizable wih eigebasis { v,, v } ad if we wrie S v v, he v v [ ] S S D v v i R cosisig of Noe: I is o always possible o diagoalize a marix We wa o udersad uder wha circumsaces his will be possible revised /6/7
Powers of a marix: If a marix is diagoalizable, we ca wrie [ ] marix S Therefore have SDS ad D Example: For he marix ( SDS )( SDS ) ( SDS ) SD S where, if, calculae for ay (posiive ieger) S S D for some chage of basis D, we ll Soluio: For his marix we foud ha S S D where S, 4 S, ad D So SDS ad 4 4 4 + 4 SD S 4 4 4 + pplicaio: Markov example There are siuaios i which a fixed amou of some asse is disribued amog a umber of sies ad where some ieraed process simulaeously redisribues he amous o oher sies accordig o fixed perceages For example, suppose you had a fixed umber of beas disribued bewee wo piles, ad process simulaeously moves 5% of he beas i pile o pile (while reaiig 5% i pile ) ad moves 75% of he beas i pile o pile (while reaiig 5% i pile ) We ca describe he rasiio as follows: If x is he umber of beas i pile ad x is he umber of beas i pile, he he ew values will be ew x 5 x + 75x deermied by ew x 5 x 5x Tha is, he ew values are deermied by applyig he marix + 5 75 x 5 5 If we hik of x x as he iiial disribuio, he afer oe ieraio we ll have x x, afer wo ieraios x x x, ad so o fer ieraios he disribuio will be give by x x The abiliy o calculae powers of a marix usig eigevalues ad eigevecors grealy simplifies he aalysis 5 75 5 75 I his case, we have 5 5, I 5 5, p ( ) de[ I ] 75 5 ( )( + 5) ad he eigevalues are ad 5 These yield he eigevecors v ad v x If we bega wih ay cofiguraio x x ad expressed his i erms of he basis of eigevecors { v, v} as x cv+ cv, he we would have x cv+ cv, x cv+ cv, ec fer ieraios we would ge x cv+ cv u wih ad 5 we see ha for all ad, so eveually x cv I pracical erms, his simply meas ha he umber of beas i each pile will eveually be proporioal o he compoes of he eigevecor v For example, if we bega wih beas iiially cofigured i ay way, eveually we ll fid he umber of beas o be approachig 6 i pile ad 4 i pile revised /6/7
Example: Fid he eigevalues ad eigevecors of he marix possible, by fidig a basis cosisig of eigevecors 7 4 ad diagoalize his marix, if 4 Soluio: efore geig sared, oe ha he colum of s meas ha e, so e is acually a eigevecors wih eigevalue Ideed, he kerel of ay marix is jus he eigespace E 7 4 I, so p ( ) de[ I ] ( )( + ) + (4 ) ( )( + ) 4 + This yields hree disic, real eigevalues, ad (Order does maer, bu be cosise) x 7 4 x v 4 4 x 7 4 x v 4 x x 4 x 7 4 x v 4 x Oce agai, we were foruae o be able o produce a basis of eigevecors { v, v, v} Theorem: Eigevecors correspodig o disic eigevalues are liearly idepede Proof: We prove his fac usig a iducive argume i which each successive sep uses he resul of he previous sep For a fiie se of eigevalues, here will be a fiie umber of seps () If here is jus oe eigevalue, he here mus be a correspodig ozero eigevecor v This is a liearly idepede se () Suppose here are wo disic eigevalues wih correspodig eigevecors { v, v } We wa o show ha hese mus ecessarily be liearly idepede To his ed, le cv+ cv If we muliply by he marix, we ge ( cv + cv) cv+ cv cv + cv ( ) The origial relaio cv+ cv gives ha cv cv, so c v + ( c v ) c ( ) v ecause ad v, we mus have c u herefore c v, so ecessarily c Therefore { v, v } are liearly idepede () Suppose,, are disic eigevalues (hece,, ), wih correspodig eigevecors { v, v, v } Oce agai, we wrie cv+ cv + cv Muliplicaio by gives cv+ cv + cv cv + cv + cv, ad he origial relaio allows us o solve for cv cv cv Subsiuio gives cv+ cv + ( cv cv) c( ) v+ c( ) v The previous sep esablished he liear idepedece of { v, v }, so ecessarily c( ) ad 4 revised /6/7
c ( ) ecause he eigevalues are all disic, his implies ha c ad c Therefore c v, so c as well So { v, v, v } are liearly idepede The argume coiues i he same fashio so ha if, are disic eigevalues wih correspodig eigevecors { v,, v k }, hese mus be liearly idepede Corollary: If is a marix wih disic, real eigevalues, he is diagoalizable Proof: If he roos of he h degree characerisic polyomial are,, each will yield a correspodig eigevecor so we ll have a collecio { v,, v } of liearly idepede eigevecors This will cosiue a basis for R, so he marix will be diagoalizable Noe: This meas ha for a marix o fail o be diagoalizable, is specrum mus coai eiher repeaed eigevalues, complex eigevalues, or possibly boh However, i is quie possible for a marix wih repeaed eigevalues o sill be diagoalizable The bes example is he ideiy marix which has oly he eigevalue bu his eigevalue has algebraic mulipliciy The ideiy marix is clearly diagoalizable because i s already diagoal! ll vecors are eigevecors of he ideiy marix Example: If we compare he hree marices,, ad C, we ll see ha hey each have he same characerisic polyomial p( ) ( ), so hey each have jus he oe eigevalue wih algebraic mulipliciy However, a quick calculaio wih each of hese marices reveals ha he geomeric mulipliciy of is (every vecor is a eigevecor), he geomeric mulipliciy of is, ad he geomeric mulipliciy of C is Neiher marix or marix C is diagoalizable, k, Noes by Rober Wiers 5 revised /6/7