defines eigenvectors and associated eigenvalues whenever there are nonzero solutions ( 0

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1 Chper 7. Inroduion In his hper we ll explore eigeneors nd eigenlues from geomeri perspeies, lern how o use MATLAB o lgerilly idenify hem, nd ulimely see how hese noions re fmously pplied o he digonlizion of mries resul iself h hs myrid of ppliions. Throughou, we ll look exmples from he field of dynmil sysems o proide he moiion for desiring digonl form of mrix. We ll work wih he m-file eigshow nd he following MATLAB ommnds: eig.. The Defining Equion The innoen looking equion Ax λx defines eigeneors nd ssoied eigenlues wheneer here re nonzero soluions ( x ) o his equion (rell h λ n e zero). Geomerilly his equion sys h if you find eor x suh h x nd Ax re prllel, hen x is n eigeneor of A. By lling he m-file eigshow from he Commnd Window, you ll lunh progrm h displys uni eor, x, nd is imge, Ax, under mulipliion y he mrix A. The mrix A is shown in he pull-down menu he op of he new window. Cliking on he Figure window nd holding he mouse uon down will llow you o drg x round he uni irle. Whereer you n disoer x prllel o is imge (he eors will urn red), you will he found n eigeneor. Go hed nd experimen wih oher mries from he pull-down menu. I s lso possile o lod x mrix ino eigshow. For exmple, >> A [ ; 4]; >> eigshow(a) This new mrix A should pper in he pull-down menu now.. 7. # Use eigshow o esime he eigeneors. Lod he mrix from he oyoe nd rodrunner exmple (in Seion 7. of he ex) ino eigshow nd ompre he eigeneors you find o he eors menioned in Cses nd found on pges Noie he eigeneors in MATLAB will e normlized.

2 5. Repe he work in Cse on pge 94 wih x (Wh does his eor men for his ppliion?) Show how o use MATLAB o deermine he oordines nd. Gie formuls for () nd r(). d. 7. #5. Algerilly Deermining Eigenlues for Mrix Trying o sole he defining equion, Ax λx, for non-zero eors firs leds us o disoering he eigenlues of he mrix A. Begin y puing ll your x s on one side Ax λ x de( A λi). ( A λi) x. We know his equion hs non-zero soluions iff Noes: We ll de( A λi) he hrerisi polynomil, nd de( A λi) he hrerisi equion. You my lso see he hrerisi polynomil wrien de(λ I - A). Some people prefer his ls form s i gurnees posiie leding oeffiien, u he sme people py higher prie ler on when ompuing eigeneors. Oserion: The roos of he hrerisi equion re he eigenlues of A. (Good ime o puse his is ig del.) Cerinly, one n use MATLAB s sole ommnd o ge he roos of de( A λi). For exmple, ry enering he following in MATLAB >> syms lmd; >> A [.86.8; -..4]; >> sole(de(a - lmd*eye())) MATLAB Noe: sole(polynomil) will proide he roos of he polynomil, h is, sole(polynomil) soles he equion polynomil. Alhough his isn oo umersome, MATLAB hs n lerne mehod for geing eigenlues of mrix: he eig ommnd. Rememer, his now mens eigeneors.

3 . Wih he mrix A oe, ener eig(a) in MATLAB. Try lso rs(eig(a)). Chek ou nd ompre he wy MATLAB sores he nswer for sole(), eig(), nd rs(eig()). MATLAB Wrning: If he eigenlues re no rionl numers, MATLAB will omi pproximing frions. For exmple, rs(sqr()) yields 9/985 (his grees wih ou o 6 deiml ples). You n see he differenes for ns in he Workspe window.. 7. #7 (My skip skehing he rjeories.) 4. Algerilly Finding he Eigeneors of Mrix A λ, plug in he One you he he eigenlues for mrix, you n go k o ( I) x ul eigenlues for nd sole eh of he differen homogeneous sysems for he differen s. For exmple, using he eigenlue. from he oyoe nd rodrunner exmple, we This sysem hs soluions. Any eor in his soluion spe seres s n eigeneor for.. he ( A λ I) x x x 4. Use he sheme jus shown o idenify n eigeneor for.9 from he oyoe nd rodrunner exmple. Now, you n go round lking ou he ker(a.i), u you ll sound relly sophisied lling his kernel he eigenspe ssoied wih.. The dimension of n eigenspe ssoied wih, lled he geomeri mulipliiy of, is simply he dimension of he ssoied kernel. The ulime eigen-word: If you n find n linerly independen eigeneors for n nxn mrix A, hen we sy h here exiss n eigensis for A. MATLAB proides wy o ge eigeneors nd eigenlues in one ommnd. By yping [P,D] eig(a) in he ommnd window, MATLAB will produe mrix P whose olumns re of eigeneors of A h orrespond olumn-wise o eigenlues displyed in he digonl mrix D.

4 Le s reisi Exmple 5 on pge o see wh numers MATLAB would produe for his prolem. >> A [.95.6;.8 ;.5 ]; >> [P,D] eig(a) P D The olumns of P don look like he eors used in he eigensis on pge, u hey re equilen. MATLAB produes eigeneors of uni lengh. Howeer, le s proeed 75 wih wh we he nd express he iniil se eor x s liner ominion of our hree eigeneors: >> [75; ; ]; >> x in(p)*; >> x(); >> x(); >> x(); >> p P(:,); >> p P(:,); >> p P(:,); >> % Chek h x *p + *p + *p

5 x( ) A x A A A A Bu eh A imes eigeneor n e repled wih λ imes eigeneor, so x ( ) A x 67.84( ) (.6) (.4) This yields j( ) m( ) ( ) (.7454) 9.878(.6) (.499) 67.84(.4) (.98) (.596) 9.878(.6) (.6656) 67.84(.4) (.596) (.98) 9.878(.6) (.5547) 67.84(.4) (.7454) And in he long run, j 5, m 4, nd. Admiedly he numers here ren nerly s nie s hose in he ex, u in his exmple we didn ines ny ime in rying o find ineger represenions for he eigeneors #4 5. Digonlizion In Exmple 5 on pge i firs ppers h we need n expression for A (MATLAB n gie you generl expression for his mrix!). For his exmple howeer, lile leerness ws used o ge round his prolem he iniil se eor ws deomposed ino liner ominion of eigeneors, hen he eigen-relionships llowed us o wrie x( ) A x A ( + + ) A + A + A λ + λ + λ

6 In he sene of n iniil se eor (or regrdless of he iniil se eor), n you sill nlyze he ehior of dynmil sysem? In oher words, n n expression for A e disoered? To egin nswering his quesion, we urn your enion o digonl mries. Digonl mries he few inriguing ris: Powers of digonl mrix re snp o ompue nd digonl mries ommue wih eh oher, u no wih generl mries. 5. Try ompuing f e d f e d Suppose n nxn mrix A hs n eigensis ( ) n,,, wih i i A i λ nd onsider [ ] [ ] [ ] n n n n A A A A λ λ λ Bsed on your oserions in prolem 5, ry foring he ls mrix on he righ ino wo squre mries. One of your mries will e S, ll he oher D. 5. Coninuing from prolem 5, he mrix [ ] n S is inerile, why? Use he equion AS (your forizion) o figure AS S. Now, if D is digonl mrix nd D AS S, i doesn ke muh o ge o A. (Imgine if D ommued wih ll ompile mries; wh would he o e rue ou A?) Noe lso h ( ) ( )( )( ) ( ) ( ) DS S D S S SD S A

7 Summry: If mrix A hs n eigensis, we n ge digonl form ssoied wih A, nd ge n expression for A y exploiing he simple lulion of ompuing powers of digonl mrix. For exmple, reisiing Exmple 5 on pge, he mrix digonlized o D y he mrix V, using MATLAB:.95.6 A.8 n e.5 >> [P,D] eig(a) P D As we expe, he min digonl of D shows he eigenlues orresponding, olumn-yolumn, o he eigeneors h mke up he olumns of V And he h power of A n e expressed s A VD V. Here is one wy o ge MATLAB o rry ou his lulion. >> syms >> D [^ ; (-.6)^ ; (-.4)^] hen >> V*D*in(V) ns (* los of numers ry i! *)

8 5 6 Bu you n see from he disply h lim A 8 48 nd herefore h lim x ( ) j lim m ( ) j 5 6 j j + 5m + 6 ( ) lim A m 8 48 m 8 j + m ( ) j + 5m + 4 j 75 Rell h m in he originl exmple. Using h iniil se eor gien in ( 75) + 5( ) + 6( ) 5 he exmple does in f gie ( ) ( ) ( ) ( ) ( ) ( ) s efore Bu now h we he more generl expression for A, we n more esily nlyze he ehior of he sysem wih differen iniil se eors.. 5d. Wih A.5. expression for A , show how you n use MATLAB o ge n..4