EEE442-HW1 - HE 441354."! W 1 Umor Morgifl Prob....

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Unformatted text preview: HE 441354."! W 1 Umor Morgifl Prob. 1 : Let. :1 he an n :r n reel matrix. Note that the P-llfll‘ifl on R” is defined {Li-I- : H. a use. = {E I x. lo I=i where I 5.“ p it: no and J: = {1:1 ..:r:..JT. ’i'stnnds for trinispme. We also define = mean-H .19.; Mm note tlml. ||::.'||2 = wit-z. The iinineedl norm of :1 is defined es: IIAJ'H Innis = sup —” = “P r;[l “I'll: .IIIiiII:I' Axflp Let Lhn unflrix A be git-nu by its nt'nJ'IjJuneuts nu. where I is the row index and j is I.In.: umlumn index. I’rove the following .- i : Ilfiilll-1 = unis: EL] nu {113113. eoiumn sum]. ii : “Aillm =11]an 3;: |o.,-_,- ]. (max. row sum}. M iii : “All” 2 Jam: where 1mm is the mnxinsuln eigenvalue of A: A. Prob. 2 : J'i. n x In real matrix A is milled a. stochastic Iuntrix if all entries ure nomnegnhive {i.e. rig-I,- :=- Li} and the sum of entries in each row is 1 [Le ELL 6,] = 1 for all For any rnunher ,8 such that ii <6. ,3 <: 11 prove tine; fie! is a contraction auni I —,(L»-i is uonsiugulsu [I is the identity mslrix}. Prob. 3 : Let the matrix A be as in Problem 2, except that instead of the sum of all entries in each row is i, this time the sum of all entries in sash column is 1. {Le E1921 eu- = l for all 3']. For any number ,8 such that H <1 i3 :2 l1 prove that $21 is a. contraction and I u 33A is Ilonsingulzn' (I is the identitfrr matrix}. Prob. 4 : Consider the feedback configuration shown helm-r. Here 9551-] : R —r R. is n differenr Liable function and It: is a. real number. Find a constraint on :35 and it: such that thissystern hes a unique solution for any,r input u. [Hint .' Toke deritrstiue, use contract-ion. mopping E of +G Graffiti ‘3‘ #. rfk ——-—— Efi mug—n mu 3 RI If an can; - q”... flnzl {qmzl+_ .lq'n-In\ ’0 LEE!" '2': “‘12. j :4: GET Q11 _ at“ 3 H1: fizlxlnkfluxfrnmwg" In an: 49-1-1 ill-1n limit—n QHLILi-Jfinfi'fi? f!) -' “liq-inf fxzf+ {fury-r} fiflxflf :- {qfl if!" fiiflzlf' "qrnwnll 4 rq2|11*1r;21*"*hm1":* +l'lfinl'1f'*afllzi+ 14"?!1‘"! E:- ("I‘a'llll'Ill rqfll“' Haunt) [Ill 1 “REM Hzflhf 'tinhfljlx;1l+' 1- {|fi!n[*lqzv1ll"" '1 “unljufl'l 4- a Kmx : WHY-1: Knlfi, _~ Lin}; "- mm‘ Z “if—#115? 1: ‘=' r 1-: Kilrerr-l k-iixzi‘F ""' Kai-Tall 5: Km "'1 thi—rL'X1I-‘I- 1-.‘31n'lf -; Han-g "1‘14 :5 KM '5? iflqflf ‘3 kflil % a" f? R Jr mm, 7;: 5119:» fire éfimUfi t fisfiumg Thu-F Ji-br aw. Jay: M=R {‘1‘ b3“:- chmfl— lib-1". '15:: “#51 ,:¢. '35:: inc... fan-a) éi) {Enema}: w; 411,1 ,ng , _. :2an §§2er= Max ‘Elllth'yl-I 46?,fl'x'mll' F Minna 4 {ah-1ran , ffi'nrgflfu +ahnwfll?‘ i :5. mar-z firéfluflmh. “fluflfinflj (Iflxfhrlfaflrfflhflxdfl’ angina,“ _ -1|' féMfiflfllf L 314+ f'xgl 5;. Hum mi ‘ > s v 1 ? $95!!!“ 5; Max. femur-r +r'fih11 “I‘m , .- j {Jump .nam‘s. III“; a L': I'laflllll" 4"Ilfi'ni 5' Z “NJ; I Jrr 2,», 31'??me 5* mar LHJLL J LHZ' mth ’1 flrufi f. Jr-l Tar show #516 EfiMaUF-j _. [4.1L Lm L [rm dig.» a.“ M .i. {Linen-E 1: (“Sprain-t]: gqcnngh}_ J ${dtaflfljj nfi HINGE-.- L J ." H TR 53’ “q”- " [ Nari-“quf-r 41Mrmir ‘5- 1"; “AR-IL“: fqm|f+ "r WIWWI". 4: b") Wang. t L”: g... 11m Maya..— 1 #7:!“ 41 , 3') flat?“ n by," = M L 55:31! I z ‘I' {n.‘J MW; : a:ch FAME : 2U? 251)“, we. need m. {fab-{5 {koa— Lifizu- an an «Age/mum '3; 9% are reu‘t m mafia”: In} A“ awn-wth Ft HTTP.- W‘L‘ “Wag-mi L-EA' 2:4 1. .2 :3“ Illa! "‘14 PE fligfl'suid-JFE 'g Ill-“FT”: l31M“ m mrré’h-P'MJ’E afuuaa-Luex .'-¢_. F1793; : 7*: 34 ~=*‘ J” T I SIM” 3.1 gal :43 Wind ELM {1.1 *1: HI Lav} dL‘gL4w fining“ {ml-P.) 2 =§I TIT-“2|: KT?! :- Dfiz-I hill-r” —I 1.1,} 1 T T T afflxflz'xnflfi a: 1: {d,1.3,4. +Mfll'fifl‘l 1 _-: fl,fl|+tx;':}2+ -f finlfl.‘ m nwm mat. 9.511. '2. :7: #42”; 5 "Jul {a}.qu 4:19:11 : '1”; wars 2 Hmn‘L 1 _ F“: NHHI ; MM ——l- 5" TH '5 Ila“; é— V21”: uw’ ————— 2 Eflfl by! Law! EE‘fl-fluj‘jj L£+ CIMF 9M {M 51H - ' :- 1 T- C. WG‘I‘C "1 a“ :55 1TATH1I: “gm”?— : flux 13.1%”: firm}! -1 JEA Ly +1.41. Mom» 1.; "9:er1 :33 IMII1= «Amrnm‘a fl.) Lu u; them? at) mrm (a; hall-ram HAL. :- Mr. “w WM=L :9 1599"“ : (311911“:941 EFL-.fiafl m A.an $.53!“ ugh-*5 or: filfih— jag-aha) = flfsflf’x-g'ltm z; mle Madam 5 [3 man“. aha: {iii a3; {:53 55. a Caa’Fr‘ar'H'a-«n To W 1" I}. _..._.E UM, wnfmflCfiflh HEEL-H “hi-'4’ '1' [31“ u'5. Ma'r 11-.th . TM»: m; {Mpg-l1 '31 1:}0 as!» "Th-4' [I- rm)?! Tu «3‘1 “3:: EH”): ‘35! HR?!“ .1: EPA-alum 43 {3 H12.qu HQ‘HM £3114 “film. 1'5. W34“ Pchallofi Haw»; 1L": - m; 4:. a Cnn'fr‘cm— uni-kt'J-W‘fifij {In 3% MFJR. fivrkn chm “Ric: (WHM'I- flu-:1 Eff.“ 1 _ {girl I“; Taber'ku 3} Sam: 9-. z" 9.1+ TL); 4-?“ chm-we 1' MIN. Mg: My a,“ 5m :1 ,mmg HIP-filth: {mmufip‘q T'Hfi rfle’a‘t 1.5 5-"vLLar .4) arm—ifi Ia: ¢l£3:¢(4.‘1~33 ‘23 _ l "Dagw Fifi): 43fu—13‘5 =1) \sziI-j] ? Wm It: is: a “pihr; P451“. sé FI‘H_ Such. ,9 unfit-1W WE] Ell-afi- ;{_ 1:4“. I'E. a Con+rGCHM. '--E. “Ha-Fm“ g whet; Ear 9M rm 75,.“4 IlFfa‘II— FIMII a; P M4,” WIN“, Fltmxtdfi I 3'6 JFz-‘I— rPFQW é) Lai- Hrmxfifiri-WI tug k H =“7‘ F512“) Haze: ,3' 94:1 1% en; fin? 3m $351.5“ Lara. fl udqvm‘? ...
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