5.4+7.1-2 - HWIO Mam 1/3 — flssiynmem‘ l0 Solu-h'ons...

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Unformatted text preview: HWIO. Mam 1/3! — flssiynmem‘ l0 Solu-h'ons Sec-Hon 5‘1 6) 11"” T 59 0t linear opErm‘vr an a vedor space. V, and le+ W be a. Tlfnvarian'l‘ subspace. of V. Claim : W «‘s 3(T) —invan‘an+ for any polynomial 30E). 2. n ff: Le+ 9(f)= a,+a,t+a2_t +---+ant 77M" 9CT)= 6191+ Q.T+ azT‘+--. + anT" 1.8+ xe W. 5?an W is T—invarfanf, +hen T00 e W)- rn [ac-r TkOC) e W V kél.t (you Can :62 Mrs 69 I'nc/uc-I-ron Or by #32 {ac—I +ha+ Me T’cych‘c. subspace 0F Vjenem-I-eol by ac) as +he ”Smalleré" T—fnvan'an-f subs/sauce 0F V can+afnmj x.) is con+arneol in W) So 301“) = aax+ G,T(ac)+0|2TL(xJ+--- 1—4,, 7‘”(x) {5 0x linear tombt'na-h‘on of elemen-fs in W, hence. 9(T)C>c) e W. 50 9(T)(w)s1 W , as desired. E (0 Final am made/ed Ami: fin- i‘he T—cyc/rc subspace. gene/“aha! by #06 vac-Inf 7?. GD v=M2x2m), T(A):(ié)A , and 2: o I) I0 W= span {25712) TLC’t’),~-} 772): (2.2 0(1):) (22) 74(2): TCTC%))- (2: 371%) _ Nfi V /‘\ '9... {dan- V n no) «W V n bu /"'\ N— p- v I A0 an ardent! basis fir Wis HM) 23)}- wa. 2 (a (My (Cayley~Hamc‘l+vn Weorem ,5);- MMn‘ces) L8+A ba an mm maf-n‘x, and It! Ht) be 4+»: charac-fEr/‘Sh‘c. [”0an ”F A- 755” FCA)=D , Hue nxn zero maf-n‘x. if LE1" T-‘Ffl-fiF" begin/en by 72L: Ax , fie. "aLA, Tfien +512 charaL-fen‘xh‘c, polynoml‘a/ 0F A :‘S +he charac—fEn‘Jf-r‘c P°’f/”°mfa! 0F T, 49 by #12 (Ag/ey-qu,‘/+pn 77;”,1‘ FLT}: 7;) “‘9 75-50 +mnfforma-h‘on. L€+/s’ be «1112 afandm-a/ Orv/freq éqxrs {or F". 72cm F(A)=f(Y.T],3) = [Fm],3 [721,3 = 0, m nxn zero ma+n‘X, M stuffed. 1E Le+ Tbfl a [mum 0/; anm‘vr on a fi‘n:‘+e-d:‘mensl‘onql vedor‘ Space V. c1533 3 m characfirrsh‘c polynomial (Tee) of T Splits, Hzerz .40 slow Hue. waraden‘sm. polynomra/ fluff) owe rash-ram a; T “it? any T—mvan‘an+ subspate W ”F V. .L’E Since. 757(k) was, we may writ-e. #‘rctfi (-1)”U:~A.)Cé-/\a)~- 66—») where nzdrmv 8y 7A‘m 5.2!, {waf) clI'W‘de5 £706) 50 Fm (t) ,3 a [grog/“d. of a #:4124224? 0;“ #:1— fad—ar; a; £705) alcove, Mme in, (t) Splints. (19 Cigar. rpm characiLE’rml-rc pofynom"“/ {7(5) 0,: TVA-+3) Hwn my naninw‘al Tim/arran+ dub/once W 07C V Confil/‘ns M a‘jenvahr 0/1 ‘7‘. ff [74%) _s};/;¥:S => Sufi) fps/15‘s (53701)) =7 Some A; I3 cm 643% Wold—0x 0F 712, WI?“ Corry/confine} ex‘gfinvec-lor v; (M ,Crw (f) oliw‘aLD/J FT (4:), 5‘0 Marga fade/r) z) y- :5 an agent/80hr of ‘7“ My, camM/Oondd'nj I 9;va A; ($5 (9) a.) W wnm‘m Me agenv’echr V; 9/9 '7”. m HWtO. :23 La» T be a linear opera-hr on 0. fim'fi—drmemianal VQC'IDI‘ space 1/, and [6+ W be a T-invan'anf subspace of V. Sup/)OSE f'haf vuvb...’ V,‘ are eigenvechrs of Tcorruponding 1'0 dlkh‘nm‘ evens/atom. £1324 If V;+Vz+“‘+VkeW' ‘f’hen V;eW VF. 2:6 For «the 65152 case, assume v,+v2. e W. Since W t: T—mvan‘anr , 7'(v,+vz ) = TLV,)+ 77.“) = Aw, + Asz e W (where A; f; +11: e/‘jenuam cor/upon din} +o 14-), Since W ’3 a Jaw/2am , A; (v, + v2) 6 W, and finer; ANN-Aw; ~Az(V.+v2) =CA,‘A,,) V,ew. Since A,¢—x\._,+hen (Aw/Ra.) 7!: 0, So _.'.____.._. (A,—A2)v, =v, c— W. W {A.~>\2.) my, (WM/2.) -v, = v; e W. Now assume +ha+ v,+--~ “’1“: e W imp/res weW, [sis k—I. (1:22) Assume. V,+ ”UH/k eW. 7Een TCV,+-~-+Vk} = A,v,+~~ + Akvk 6W as W ,‘5 7.,‘nvarl‘anf. Since W is a :ubs/aace , Au, (V,+‘“+Vk) 6W; and fiver: /\,v,+-H+Aka ’2‘». (V:+"‘+Vk) =CA.~A,,_) V. + (AL-Ah”z + 4- (AM —>\k) vb, :- W. 5I‘nce. ‘HIC ANS are oil‘s-find , (Ai'Akjny V ’5" $’<"/- Sincfi V.,...,Vk-, are eigenvéc+or5 , AD are (AI~Ak)V., J (Au-umg) Vu—l 777.2!) by our I‘naluc‘h‘on hypo”) €513, (I‘m-Mn,- e w V Is I‘Sk~l. Since (A.‘*}\k) #0, (’\"‘)‘k)":' =V;C- W V I$l'$k~1_ CAi—‘Ak) 5,-me W o‘; a nubdfiace , (v,+-.. +Vk) -— vz— V3 _. -Vk_’ I: Vk G W_ .3 V" E W V [Si‘é l<‘ WM 3 wa. @ [2+ T baa Iz‘near‘ operafor‘ on a v€c+vr space V, 4nd le+ Wu WZJ-w Wk be T-z’nvarian'l- subs/paces of I/. Prove fimv‘ W,+Wz+ “‘+Wk 1‘5 «150 a T-I‘nvara‘an'l- ”(bx/Jana 0F V. ff Le! X€W,+~-+ Wk . 50 x=w,+---+wk , for-:ome w,‘EW;. Wen T(K)= TCW,+ +Wk) = 737104,) + "“'+T(Wk) ($3 lineqrf‘fy) G W, + + Wk since. 7"(w,') e W,- {er each i as +162. W; are T-z‘nvan‘an‘l‘. .‘.T(W,+~«+Wk) g WI+"'+Wk and we know J’ha-f- 4472. guru} dam of dubA/OOILE/J 1:4 “flair: a Aruba/941418.. 72:45 W,+ ---+ Wk is ox T—invara‘qn-i- SubS/pacC of V. @ Answer : (“Mn-z £2. __ nag-H) t _, n3(n+i)(n-I) 2. [2. @ Aniwe/‘i (-1) 19”“ (t —n) Seth'onV-J @For each ma+rix A, find a basis fir each genera/gee! eigempace of LA C9”$,‘SHH’ of a umon of chisjoinf cycles of general/36d a/‘jenvec-fvrs. 7730: find- a Jordan, canom‘cal form J‘of A. (9A: ’2 -..-.- de+(A’+.I =ole+ I—t 2 = .. - - (32.) 7 ) (3 M) £121“) 6 ’ ~ -’4 ’(t-I-Iflt—H) Eigenvalues A = ~I and A: L] , each wi'f’h mulh‘pliu'+5 I wmq dim (/<-,) .- dim (Kg) = I. For A=-I: E,=N(T+I) => E,=K, (a :m =6) m = w.» WW I : i ( ’)} 0rd8r€d 501553 for -! ForA:‘4: E” = N(T"II) 2:) 531:1“! (-5 m) = (s) (2;) = H?) > WW3 V. HW‘O. 6 -= 9- ' 00 .—.— Ae+ A-tl =4 + bk 1 o O .-.- (24316-01 02 3 O > ( ) e o 24: I 0 00 3 0 o 0 3—t 0 014 3 o I -: 3—i- Eigenvalues A:Z and A=3 , each wffh mulhyah‘ully 2- 50 0“”! (K2,) = CUM (K3) =2. ForA:2.I AimCE;)'-‘LI’3=I7="‘AMCKL) :) 758 bQSI‘S 0,5 K2, is a 5/0le cycle of leaf)”: 2. . N((A-21)1)=N (2213 = 2C(é>+d<?) 590! Julian} 00:0 0 0 Ol-li ° ”' wa. 3‘ @ 11+ U 5261 ”near 0f erm’vr‘ on a fi'nl'l'e~cli‘me05f0fla/ Vet/'0” Jpace V‘ Prvvei __' uh) C N(uk+/ C. _ - @ N(U)§N(ul)$ ENC ‘. ._ ff: Le+ xe NCu“) ; k‘zi , keZ. 756/7 Uk(x)=0 / So uk+l(x)= u (uka)>.—=U(O):0 2’) xe Néukfl)‘ Since kZ/ was arbiffll’y, Wf baa/:2 NCU)§N(UZ~)Q...S MCI/(k) $N(uk+l)g ... m (b? IF ran/c (um):- amicagm“) 7C0,» some/oorl'z‘v‘ve m Hjer m, ”Len. rank (Um): rank (Mk) For any pas/five, I'm’fjer k) m, 1:; Assume, rzmk (Um)=rank(li””") for Some. rag-2+, We know Um“(l/)= mm») 2 u’”(v) Since mag“ 50 #261? dimension; Ael‘nj ezuaj.’ I‘m; lies W*’(V)=uwv). Le+ k? rm—I. 75m (uh/v) = MN” (gnaw) =u""°‘(u’"“m) = M” (1/) . 50 ukmzuk“ (V) v kzmn ”75m WWW: MHV) Vk7/m. 50 rank (um) : rank (M) V kZM. @ @ If? rank. (11“) =ran/c (Um’fl) FVSOME/aonfi'rf I‘nfejfr M, file/L Nam): NUJ") for any/00517796. I‘m‘vgfer kan. ff Ami/146 ran/C(U’”) =rzml¢ CUM“) 765’“ ”MC “462+- m by [Gaff @) aka/my»: mnék (ukfl/D 17‘ k7/m. 5y :H’ILR ank -NM/lFIy WM} mfg [Mp/{‘85 cit-may)- mums/3 (W) = a/WW)- null/19 (11“) 3) nui/mj (W) = nod/1+3 (Yuk). 7;;75 comb/med «21% 7‘25 {QuL 7‘th NCU’”) 5; N(Uk) 90m @ mm M Nam): NM") l/kzm. E HIM/(l 5 @243 Lama lmear opGFa—br on 1/5 and/81‘ A At; an 8%,,an a; 7-_ Ola/I‘M: 1-; rank (KT-AI) m): rank (aux) m“) 5, Some infeje/ m, m K} = N((T~)\jj) m) If Aezal/ KA: {kg—V3 (T—AI)P(>L)=O firjome FeZ+3 So KA = N (”FAD U N((7‘——AI) L) u . . . u N((7:./\I)’") U NéT~A£)W§+1)UH. 5u+ N{(71AI)”‘) = N((-7‘~AJ) k) Wczm by 0. So K3 = NCFAI)U NUT—ADE) u u N((7‘~AI)"’) 8H4» MHz—AI) 9.: N((7-AI)1) Q S N (UM—W”) by @/ .S‘o KA :. N((T—-)\1i)’7’).~ w @953 5910’”? 7735+ for Diajon a/I‘jaéf/ffy . 1.6+ 7'56 4 //‘fl€afc}o€fa1‘vr an V w/o stalks, and lev‘ A” A;,-~,)\ k Ac Me disfind gym value; 0/7’. CMIM .' T [‘5 d/‘ajona/[jaéle :55- ”40k (T-A I) ‘-’~' mnk ((T‘AIJZ) oJ‘C (1mm tiff/7577C polynomial, For léi‘ék. BE (We‘ll be Jena-Hg A; 53/2} gar-AIM,» lich‘y) (7—?) x455ume 7‘13 almjonugaue. .2? KA: EA : Nan—AI), y/Sfék (by Conv‘o 713m?.’1) => Since, N((7‘-/U:)?)S KA, Men N((T~AI)9=N(T~AI) => nunHyafi/xzfi) - mum-79 (727ng My“) ==> rankaT—Arfi) :- rank(T—AI) 5y mnb/VMW . (67-) AMI/(me. rank (OZAIJL): rank (72);) / Vléiék -==> K) = NC'FAI)’ 1 VI $16k; 5.9M? m=/"°@ MJM know +142, ohmm‘a'sfic fojnorm‘ai 0; TJfl/b (by (or. 4D 75710319) :27 775 d/‘azyonal'gaéle. g ...
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