Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are o hs beng possble We ll also look a some algebrac nvarans of a lnear ransformaon, namely s race and deermnan, and relae hese o he egenvalues of a marx represenng he ransformaon We ll also dscuss wha he presence of a complex conjugae par of egenvalues means n erms of nvaran subspaces and how he ransformaon acs whn such a subspace Summary of resuls If A s an n n marx, we call a vecor v an egenvecor of A f T ( v = Av = λv for some scalar λ Ths scalar λ s called he egenvalue assocaed wh he egenvecor The exsence of an egenvecor depends upon wheher here are any soluons o he equaon ( λi Av =, and hs s only possble f he marx λi A s no nverble A necessary and suffcen condon for hs s ha pa( λ = de( λi0 A = 0 Ths wll always be an nh degree polynomal n λ called he characersc polynomal of A So λi A wll have a nonrval kernel f and only f λ s a roo of hs characersc polynomal The egenvalues are herefore he roos of he characersc polynomal The se of all egenvalues of a marx A s called he specrum of A y he Fundamenal Theorem of Algebra, hs can always be facored as a produc of lnear facors and rreducble quadrac facors The lnear facors yeld real roos, and he rreducble quadrac facors yeld (by he quadrac formula complex conjugae pars of roos If an egenvalue λ occurs as a repeaed roo of he characersc polynomal of A, we refer o he mulplcy of he roo as he algebrac mulplcy of he egenvalue For any egenvalue λ of A, he subspace ker( λi A s called he egenspace of λ, or E λ The geomerc mulplcy s dm[ker( λi A ], e he number of lnearly ndependen egenvecors assocaed wh hs egenvalue n If, for an n n marx A, we are able o consruc a bass = { v,, v n } for all of R conssng of egenvecors, we call hs an egenbass and say ha he marx s dagonalzable If A s dagonalzable wh Av = λv 0 egenbass = { v,, v n } and S= v 0 vn,, hen [ A] = S AS = D = Avn = λ nvn 0 λn If a marx A s dagonalzable and we wre [ A] = S AS = D for some change of bass marx S, hen 0 A = SDS and A = ( SDS ( SDS ( SDS = SD S where D = 0 λn We proved ha egenvecors correspondng o dsnc egenvalues are lnearly ndependen whch yelded he corollary ha f A s an n n marx wh dsnc, real egenvalues, hen A mus be dagonalzable Ths also means ha for a marx A o fal o be dagonalzable, s specrum mus conan eher repeaed egenvalues, complex egenvalues, or possbly boh I s possble for a marx wh repeaed egenvalues o sll be dagonalzable In he case where all of he egenvalues of a marx are real bu wh some mulplcy, as long as GM ( λ = AM ( λ for each egenvalue λ (ha s, he geomerc and algebrac mulplces are he same, he marx wll sll be dagonalzable So he only obsrucons o beng able o dagonalze an n n marx are he exsence of complex egenvalues or havng a repeaed egenvalue where s geomerc mulplcy s srcly less han s algebrac mulplcy revsed 4/9/05
Example: Fnd he egenvalues and egenvecors for he marx A = and deermne wheher hs 4 6 marx s dagonalzable Fnd an expresson for Ax 0 for he vecor x 0 = + Soluon: We frs wre λi A = λ+ The characersc polynomal s hen: λ 4 p ( λ = de( λi0 A = ( λ+ ( λ 0λ+ 0(λ0 0( 0 λ = λ 0λ+ = ( λ0 ( λ+ = 0 A Ths gves he egenvalues λ = wh algebrac mulplcy, and λ = wh algebrac mulplcy 0 0 0 0 Takng λ =, we seek egenvecors: 0 0 0 0 0 0 ecause hs has rank, s kernel wll 0 0 0 0 0 0 have dmenson, so he geomerc mulplcy wll be For egenvecors we can parameerze he kernel as x = s+ 0 0 x = s, s 0 x = +, so he egenspace s spanned by, 0 Takng λ =, we seek x = 0 0 0 0 0 0 0 0 egenvecors: 0 0 0 0 0 0 Ths has rank and he geomerc mulplcy s For 06 0 0 0 0 0 x = egenvecors we can parameerze he kernel as x =, x =, so he egenspace s spanned by x = 0 Taken ogeher, =, 0, s a bass of egenvecors, so he marx s dagonalzable even hough 0 0 0 0 here was a repeaed egenvalue If we le = 0 0 S, hen we can calculae S = 0 0 and 0 0 0 0 0 0 0 S AS = = D We can hen wre A = SDS and A = SD S, so 0 0 0 A x 0 0 0 0 0 6 0 ( 0 04 + 5( 0 = SD S x = 0 0 0 0 0 = 0 ( 0 0 = 04 + 5( 0 0 0 0 ( 0 0 ( 5 + 5( 0 0 0 0 Proposon: If wo marces A and are smlar, e f = S AS for some nverble marx S, hen hey have he same characersc polynomal and herefore he same egenvalues wh he same algebrac mulplces Proof: p ( λ = de( λi = de( λi S AS = de( S λis S AS = de[ S ( λi A S ] = de( S de( λi Ade( S = de( λi A = p A ( λ revsed 4/9/05
Proposon: If wo marces A and are smlar, hen hey have he same egenvalues wh he same algebrac mulplces and he same geomerc mulplces Proof: λi = λi S AS = S λis S AS = S ( λi A S Therefore, f v ker( λi, we ll have ( λi v =, so S ( λi A Sv = ( λi A Sv = Sv ker( λi A Smlarly any w ker( λi A wll gve S w ker( λi ecause S s nverble hs correspondence s an somorphsm, so he subspaces mus have he same dmenson Therefore, f λ s an egenvalue of A (and hence, hen s geomerc mulplcy wll be he same for boh A and The egenvecors, however, wll no be he same Trace and deermnan If wo marces A and are smlar, we have already shown ha de A= de A homework exercse also shows ha race A= race, where for an n n marx A s race s he sum of s dagonal elemens he case of a dagonalzable marx, hs means ha s race mus be he sum of s egenvalues, and s deermnan mus be he produc of s egenvalues In fac, hese saemens are rue generally: Theorem: If A s any n n marx wh egenvalues { λ,, λn}, ncludng any repeaed egenvalues, hen r( A = λ + + λn and de( A = λ λn Proof: Ths s easy o show for marces The general case s lef as an exercse Complex egenvalues If an n n marx A has any complex egenvalues, we wll no be able o produce any real egenvecors However, as n he case of real egenvalues, he algebra can sll formally proceed n he same manner Our goal wll be o produce bass vecors assocaed wh any complex egenvalues such ha he marx relave o hs bass has a smple, f no dagonal, canoncal form In order o do hs, we have o emporarly wander off no he world of complex numbers, complex egenvalues, and complex egenvecors You should no aemp o vsualze a vecor whose componens are complex numbers Ths s merely an algebracally conssen exenson of he dea of real vecors and real marces where all he rules of lnear algebra are sll n effec Ths emporary excurson wll yeld real vecors relave o whch he marx acs n an easy-o-descrbe fashon, namely as a roaon-dlaon, e roaes vecors n a -dmensonal (nvaran subspace and scales hem by he modulus of he complex egenvalue ascs of complex numbers Im Frs, we need a few basc defnons assocaed wh complex numbers A complex number z = x + y, where = - can be vewed n vecor-lke erms n he complex plane as shown n hs dagram o z = x + y he rgh We defne: z y = Im(z modulus(z = mod(z = z = x + y y argumen(z = arg(z = an ( x = We add complex numbers by addng her respecve real and magnary pars, n much he same way as vecor addon was defned We mulply complex numbers va he dsrbuve law and he fac ha = - For example: If we noe ha ( + ( 4 = 8 = ( + 8 + ( = 5 4 x= z cos and y = z sn, hen we can wre x = Re(z z = x + y = z cos + z sn = z (cos + sn = z e (polar form, usng Euler s Formula A shor calculaon shows ha when we mulply wo complex numbers, we mulply her modul and add her n = a In Re revsed 4/9/05
( + argumens Specfcally, z z = z e z e = z z e You may wan o ry hs ou wh some smple complex numbers o convnce yourself of hs fac The complex conjugae of z = x + y s defned o be z = x y In he complex plane, z and z are reflecons of each oher across he real axs I s no hard o show ha z z = z z 4 The Fundamenal Theorem of Algebra guaranees ha, a leas n heory, any polynomal of degree n can be facored no n lnear facors and wll herefore produce n roos, ncludng mulplcy Some roos may have mulplcy greaer han and some of he roos may be complex I s also he case ha for a polynomal wh all real coeffcens, any complex roos wll necessarly occur n complex conjugae pars λ and λ (hs follows from he quadrac formula, resulng from an rreducble quadrac facor Le A be a marx whch has a complex conjugae par of egenvalues λ and λ We can proceed jus as n he case of real egenvalues and fnd a complex vecor v such ha ( λi Av = The componens of such a vecor v wll have complex numbers for s componens If we wre λ = a + b, and decompose v no s real and magnary vecor componens as v = x+ y (where x and y and real vecors, we can calculae ha: ( Av = Ax + Ay = λv = ( a + b( x + y = ( ax by + ( bx + ay If we defne he vecor vˆ = x y and use he easy-o-prove fac ha for a marx A wh all real enres we ll have Av=Av= ˆ λv= λv, ˆ we see ha vˆ = x y wll also be an egenvecor wh egenvalue λ, and: ( Av = Av ˆ = Ax Ay = λ vˆ = ( a b( x y = ( ax by ( bx + ay The rue value of hs excurson no he world of complex numbers and complex vecors s seen when we add and subrac equaon ( and ( We ge: Ax = ( ax by Ay = ( bx+ ay Afer cancellaon of he facors of and n he respecve equaons and rearrangng, we ge: Ay = ay + bx Ax = by + ax Noe ha we are now back n he real world : all vecors and scalars n he above equaons are real If we use he wo vecors y and x as bass vecors assocaed wh he wo complex conjugae egenvalues, grouped ogeher n he full bass = { v,, v n }, we ll produce a (Jordan block n he marx [ A] of he form: where a b a b a b a b cos sn a b + + o o b a = + b a = sn cos = R a + b a + b R s he roaon marx correspondng o he angle = arg( λ In oher words, he Jordan block assocaed wh he bass vecors { yx, } s a roaon-dlaon marx where he angle of roaon s he same as he argumen of he complex egenvalue and where he scalng facor s jus he modulus (magnude of he complex egenvalue Agan, he very naure of he complex egenvalues ells us much abou he way he marx acs, a leas f we choose he rgh bass wh whch o vew hngs 0 0 Example: Consder he marx A = 0 You ll recognze hs as he marx correspondng o counerclockwse roaon n he plane hrough an angle of 90 The characersc polynomal s λ + = 0, wh revsed 4/9/05
complex egenvalues λ = ± Noe ha wh λ = +, we have arg( 90 λ = and modulus( o = o = The precedng dscusson says ha hs marx s smlar o a roaon-dlaon marx whch does no scalng and whch roaes by an angle of 90 u hs should come as no surprse a all The gven marx s already n he form of exacly hs roaon-dlaon marx, e Jordan form Example: Consder he marx A = We have λi A = λ, and he characersc polynomal s λ 04λ+ 7= 0 Ths yeld he wo egenvalues λ = + and λ = α If we subsue λ = + no λi A = λ, we ge ha f v = β s o be an egenvecor, we α 0 mus have = 0 β 0 Ths means ha ( α + β = 0 (The second equaon s redundan, even hough hs mgh no mmedaely appear o be he case One choce for α and β s α =, β = Ths 0 gves us he complex egenvecor 0 v= = + = + 0 0 0 x y Usng = { yx, } =, 0 as a 0 0 cos sn bass, and callng S = 0 0, we have ha [ A] = S AS = = 7 = o R sn cos where 0 R s he roaon marx correspondng o he angle = arg( λ = an ( 4089 If we have need o consder he powers ([ ] λ λ = = = A S A S S R S SR S o roaon hrough he angle and scalng by he facor A= SA S and A for any posve neger power, we ll have ha [ ] In oher words, excep for he change of bass, A corresponds We ll look a a few more dealed examples nex me We ll also ake up he dscusson of he Specral Theorem λ Noes by Rober Wners 5 revsed 4/9/05