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EEE442-HW1 - HE 441354."!W 1Umor MorgiProb. 1...

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EEE442-HW1 - HE 441354."!W 1Umor MorgiProb. 1...

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Unformatted text preview: HE 441354."!W 1Umor MorgiProb. 1 : Let. :1 he an n :r n reel matrix. Note that the P-llli on R is defined {Li-I- :H. ause. = {E I x. loI=iwhere I 5. p it: no and J: = {1:1 ..:r:..JT. i'stnnds for trinispme. We also dene = mean-H .19.; Mm note tlml. ||::.'||2 = wit-z. The iinineedl norm of :1 is dened es:IIAJ'HInnis = sup = Pr;[l I'll: .IIIiiII:I' AxpLet Lhn unrix A be git-nu by its nt'nJ'IjJuneuts nu. where I is the row index and j is I.In.: umlumnindex. Irove the following .-i : Ililll-1 = unis: EL] nu {113113. eoiumn sum].ii : Aillm =11]an 3;: |o.,-_,- ]. (max. row sum}. Miii : 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; e! is a contraction auni I ,(L-i is uonsiugulsu [I is the identitymslrix}.Prob. 3 : Let the matrix A be as in Problem 2, except that instead of the sum of all entriesin 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 conguration shown helm-r. Here 9551-] : R r R. is n differenrLiable function and It: is a. real number. Find a constraint on :35 and it: such that thissystern hes aunique solution for any,r input u. [Hint .' Toke deritrstiue, use contract-ion. mopping Eof +G Grafti 3#. rfk - E mugn mu 3RI If an can; - q... nzl {qmzl+_ .lq'n-In\0 LEE!" '2': 12. j :4: GET Q11 _ at 3 H1: zlxlnkuxfrnmwg"In an: 49-1-1 ill-1n limitn QHLILi-Jn'?f!) -' liq-inf fxzf+ {fury-r}xf :- {q if!" iflzlf' "qrnwnll 4 rq2|11*1r;21*"*hm1":* +l'lnl'1f'*alzi+ 14"?!1"!E:- ("Ia'llll'Ill rql' Haunt) [Ill 1 REM Hzhf 'tinhjlx;1l+' 1- {|!n[*lqzv1ll"" '1 unlju'l 4- aKmx : WHY-1: Knl, _~ Lin}; "- mm Z if#115?1: ='r 1-: Kilrerr-l k-iixziF ""' Kai-Tall 5: Km "'1 thirL'X1I-I- 1-.31n'lf-; Han-g "114:5 KM '5? iqf 3 kil % a" f? R Jrmm, 7;: 5119: re mU t sumg Thu-F Ji-br aw. Jay: M=R{1 b3:-chm lib-1". '15:: #51 ,:. '35:: inc... fan-a) i) {Enema}: w; 411,1 ,ng , _. :2an2er= Max Elllth'yl-I 46?,'x'mll' F Minna 4 {ah-1ran , f'nrgfu +ahnwl?i:5. mar-z rumh. unj (Ixfhrlfarfhxd angina, _ -1|' fMlf L 314+ f'xgl 5;. Hum mi > s v 1 ?$95!!! 5; Max. femur-r +r'h11 Im , .- j {Jump .nams. III;a L': I'laflllll" 4"Il'ni 5' Z NJ;I Jrr2,, 31'??me 5* mar LHJLL J LHZ' mth1ru f. Jr-lTar show #516 EMaUF-j _. [4.1L Lm L [rm dig. a. M.i.{Linen-E 1: (Sprain-t]: gqcnngh}_ J ${dtajj n HINGE-.- LJ." H TR53 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') at? n by," = M L 55:31!I z I'{n.J MW; : a:ch FAME : 2U? 251),we. need m. {fab-{5 {koa Lizu-an an Age/mum '3; 9% are reut m maa:In} A awn-wth Ft HTTP.- WL Wag-miL-EA' 2:4 1. .2 :3 Illa! "14 PE ig'suid-JFE 'g Ill-FT: l31Mm mrrh-P'MJE afuuaa-Luex .'-_.F1793; : 7*: 34 ~=* JT ISIM 3.1 gal :43 WindELM{1.1 *1: HI Lav} dLgL4w ning {ml-P.)2=I TIT-2|: KT?! :- Dz-I hill-r I 1.1,}1 T T Tafxz'xn a: 1: {d,1.3,4. +Ml'l1_-: ,|+tx;':}2+ -f nl.m nwm mat. 9.511.'2.:7: #42; 5 "Jul {a}.qu 4:19:11 : '1; wars2 HmnL 1 _F: NHHI ; MM l- 5" TH '5 Ila; V21:uw 2E by! Law! EE-ujjj L+ CIMF 9M {M 51H -' :- 1 T-C. WGIC "1 a :55 1TATH1I: gm? : ux 13.1%: rm}! -1 JEA Ly +1.41. Mom 1.;"9:er1:33 IMII1= Amrnma fl.) Lu u; them? at) mrm (a; hall-ramHAL. :- Mr. w WM=L:9 1599" : (311911:941EFL-.a m A.an $.53! ugh-*5 or: lh jag-aha) = fsfx-g'ltm z; mle Madam5 [3 man.aha: {iii a3; {:53 55. a CaaFrar'H'a-nTo W 1" I}. _..._.E UM, wnfmChHEEL-H hi-'4 '1' [31 u'5. Ma'r 11-.th . TM: m; {Mpg-l1'31 1:}0 as! "Th-4'[I- rm)?! Tu 31 3:: EH): 35! HR?! .1: EPA-alum 43 {3 H12.qu HQHM3114 lm. 1'5. W34 Pchallo Haw; 1L": - m; 4:. a Cnn'frcmuni-kt'J-Wj {In 3% MFJR. vrkn chm Ric: (WHM'I- flu-:1Eff. 1 _ {girl I; Taber'ku3} Sam: 9-. z" 9.1+ TL); 4-? chm-we 1' MIN.Mg: My a, 5m :1 ,mmg HIP-lth: {mmupqT'H reat 1.5 5-"vLLar.4) armi Ia: l3:(4.1~3323 _l"Dagw Fi): 43fu135 =1) \sziI-j] ?Wm It: is: a pihr; P451. s FIH_ Such. ,9 unt-1W WE]Ell-a- ;{_ 1:4. I'E. a Con+rGCHM. '--E.Ha-Fm g whet; Ear 9M rm75,.4IlFfaII FIMII a; P M4, WIN, Fltmxtd I3'6JFz-I rPFQW ) Lai- Hrmxfiri-WItug kH=7 F512)Haze: ,3' 94:1 1% en; n? 3m $351.5 Lara. udqvm?...
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