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CU-Boulder PHYS 3210 - Linear Algebra Part 2

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Physics 3210 Spring 2008Classical Mechanics & Mathematical Methods 2Section 5 - Linear Algebra Part 2Copyright @2008 by Joel G. Broida. All rights reserved'Linear Algebra -- Part 2eigenvalues and eigenvectorsdivision algorithmremainder theoremfactor theoremalgebraically closed fieldfundamental theorern of algebraalgebraic mu ltiplicityeigenspacecharacteristic matrixcharacteristic equationcharacteristic po lynomialdiagonalizable matrixblock matrixinvariant subspacedirect sum of maticesreducible matixgeometric multiplicitydiagonalizing normal maticesHermitian adjointHermitian and unitarv matricesnormal matixorthogonal mafixSchur canonical formunitarily similarbilinear formssymmeticquadratic formelementary mafiicescongruence transformationrank of a bilinear formsignatureSylvester's theorem (law of inertia)positive definite bilinear formoverview of diagonalizationadjoint operatorHermitian operatorI45669llt2t2l3l82l2428283237373738393942475l52546668697l72779098dt-v- gV-a.t--T-Ez)y-.- -U2.44" &11, qj-- ,e,ah,",'/-'a-) --D-aL d1"4 sy'o e,4fi;te, Jd,4ot 8-a:a 0 0-,-q-l-G)r-L!-e-kQ e<,{--vQV= ,(-Let\ a e-,1'"2{*-V llrro4-h- - Tr-:Tt-E: e-\ = ni -( e; z-gdty: ---ufuz-1o*4.".''o+v afft (:(n'fu', /-/& &u{f",--.e--1*.yJ M /'L.4'. flr;rqQte- -i tt Aice J ue ,ua lto e Ttcr-l- c'Tv = clq-z)tql-4oda1-dL-a-vL+hr*"-o --.> v'oLv+$ -o -a(r+ *)'- #. f,. =oa K= -t;sl$;-T = -J!)Ec-rzAha.^e) -*&,fot\l-> L3+/}L+tt1,2t- ,*^"2-/tzfu //^f%fr- ( pLd.,-;,/.l)*-! -rr".k^--u{ r* *-@eA a/" / o*l -s-tW,,--,tL-- trr e ?[A-2",-Lgra* -B4-L=ffifu-e-tu'---\ ; gAyu {s_ ?kl c*,1 e e3A* etL-Ae\grvt+ lCr\ _IileL4_ L=,-&-- r) 7 *_r _/lt*_+t a = La- dJ*41&r.- tz:e=t--qd*;Lc?./.e-- f: e? Gs.- a--ev*t--) . L 4ret"-Y,*,. . /* E-24, t-ffi.efAr-ts --c)7k\ +r -z-r &-fc-cLr-o-__K___4A"24-it'^ X **X *-y-"I-*@'4-\ L?, /1*%44.d4f r,rrla"(f t L-fu+14*+, d,iLf#sa,&& -fu*tu*p--lL*q,*a-l.k- ezy-e._4t, rrr.ttte ' i, a, € ? .?JqLtxl 4t* ilqQza-zue&ts de.,-3-4,4\, -,4€ 7z-1. 4cQ-4*-d]e's-Jzke)f*- -r'04+-a !-Ivf -4:L=1fu-;&-try44-.L4" hrtgu"+ 4 &ez! z n0'[0-, - n--;z4t4 , _lunt-M.eil" ryif "o,*t, ( p ; ( rz-te"r*-+-..-*-a,ry3Lh+ r-|,!-t - :,,+-U#:\-)( 3 -t-a, -€3- a.L 7; +; z-eru{h..x. C-l izft*@dE+bv+ezo4u&-*4-ML & /,*,1 fug/**;dItot e (n u "-*t1, "" ,a,ol!*Vfu*+-te/d-LilS fR rc,'n*,u* . q**l+4!L= L e"d2o-.-a .*nA(.r: --- t*l ;=--Qa)',-r-r:::*_-rs)a-$--O > Q(),- f*Jrt + - - - + c-(E- AoJu,10iAl',/rrl l"^q %, = cr\t .:*c6v(. = -o , W'Xe+3.--4o44;64-e{"",-r."* La+-@Xse + tk{.;e:*"- Y4e &?+W-A g-u YL n " o -drb-L=-ry--' - K*--l1-v_L ;- 24pr- Te- lJ u) r, ./ [.*.]/=-L-4(-r-/*---adt- 44* z{;** Le4'L 4-VJ,'L4L-7T.4a14.'.&24-ryL -,*LJs4*t a( , e L!/t- -e.J ) ; e.. 4<*"&'e'L {tn * , '{ T"q)+-u1-Ttr*==)q1_, froo--___------|sJi'tIq.4 eTieza*a . il4/1--r'i^ d i til v!-#eL i-=-!, .---rL -,- l&, z*"q-,& -le-fuU+=};jut'-^AA-U a " 4( ,ht::Idt:*l'o_ (.-- +/-_%..44_*-r{T- YU /Lb,-,,-,.'d, tU-kE- r) ;/9i ,t/ (xt -r\ = I-?4rV-sa,-\-:::/f-*!,._l-___tfu. tfu_{*__r-sil__Lr4- 14re"-4.u2.4 J-h*l.fu)fuLryF?-WL,/{_b-r) =/\-f - &-f{r+r\:P*5-fr4L4&-bt e. W -Vd-L*-rbsu;r.lTu:t-.d.,2=./s;.'.t7-*4--44*'* W -T -e-KJ4-ru- b-,urwpru +-'F r( ;d "L4+ -Vb-" a-A*-" u.i*a+.p*fi&! " YU V 0 vi-uaryZ.+-@r*[email protected]**,/-',LIe;Lft arufu ,4fuA'f,ef.re! { v L4 aA'7*^l- z.ont*-b-- fuan*-(41- &-4* /.&"\ LD.,*'k.fu----IXL f"^,'*- ry -%dlft /ce+ --/"t*&fee\J-eKv\ ca- .6 7"" ,- {&=5 c/Y,b*Agw*, st-*-&-a- teu-- r* -q ^Afr #"/h *t a-e*u .{rr,,r*,a,t A", .%4?ilq te, 4r;/-* A4 @il-eok^e. $*'fL/J* q&M!]-uu;tu"-br>\-r{_"__{6&n&@*+! e-\U)-*nL eJ&)e;LgltuEa @ 744 -ze-t,r;L4@44 ( (+ *tt"*-4* ;&4"-"**t*-*A._L e*\-fui,fufr 4 fri"etu--E-----,&'2.*t*P*,4\, lr oll(t


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