Eee442-hw1

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Unformatted text preview: HE 441354."! W 1 Umor Morgi Prob. 1 : Let. :1 he an n :r n reel matrix. Note that the P-llli 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 dene = mean-H .19.; Mm note tlml. ||::.'||2 = wit-z. The iinineedl norm of :1 is dened es: IIAJ'H Innis = sup = P r;[l I'll: .IIIiiII:I' Axp Let Lhn unrix A be git-nu by its nt'nJ'IjJuneuts nu. where I is the row index and j is I.In.: umlumn index. Irove the following .- i : Ililll-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; e! 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 conguration 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 Grafti 3 #. rfk - E mugn mu 3 RI 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- a Kmx : 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 Jr mm, 7;: 5119:» re émU t sumg Thu-F Ji-br aw. Jay: M=R {1 b3:- chm lib-1". '15:: #51 ,:¢. '35:: inc... fan-a) éi) {Enema}: w; 411,1 ,ng , _. :2an §§2er= Max Elllth'yl-I 46?,'x'mll' F Minna 4 {ah-1ran , f'nrgfu +ahnwl? i :5. mar-z réumh. unj (Ixfhrlfarfhxd angina, _ -1|' féMlf 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 Jrr 2,», 31'??me 5* mar LHJLL J LHZ' mth 1 ru f. Jr-l Tar show #516 EMaUF-j _. [4.1L Lm L [rm dig.» a. M .i. {Linen-E 1: (Sprain-t]: gqcnngh}_ J ${dtajj n 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') 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-mi L-EA' 2:4 1. .2 :3 Illa! "14 PE ig'suid-JFE 'g Ill-FT: l31M m mrréh-P'MJE afuuaa-Luex .'-¢_. F1793; : 7*: 34 ~=* J T I SIM 3.1 gal :43 Wind ELM {1.1 *1: HI Lav} dLgL4w ning {ml-P.) 2 =§I TIT-2|: KT?! :- Dz-I hill-r I 1.1,} 1 T T T afxz'xn a: 1: {d,1.3,4. +Ml'l 1 _-: ,|+tx;':}2+ -f nl. m nwm mat. 9.511. '2. :7: #42; 5 "Jul {a}.qu 4:19:11 : '1; wars 2 HmnL 1 _ F: NHHI ; MM l- 5" TH '5 Ila; é V21: uw 2 E 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-ram HAL. :- Mr. w WM=L :9 1599" : (311911:941 EFL-.a m A.an $.53! ugh-*5 or: lh jag-aha) = fsfx-g'ltm z; mle Madam 5 [3 man. aha: {iii a3; {:53 55. a CaaFrar'H'a-«n To W 1" I}. _..._.E UM, wnfmCh 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 «31 3:: EH): 35! HR?! .1: EPA-alum 43 {3 H12.qu HQHM £3114 lm. 1'5. W34 Pchallo Haw»; 1L": - m; 4:. a Cnn'frcm uni-kt'J-Wj {In 3% MFJR. vrkn 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-lth: {mmupq T'H reat 1.5 5-"vLLar .4) armi Ia: ¢l£3:¢(4.1~33 23 _ l "Dagw Fi): 43fu135 =1) \sziI-j] ? Wm It: is: a pihr; P451. sé FIH_ Such. ,9 unt-1W WE] Ell-a- ;{_ 1:4. I'E. a Con+rGCHM. '--E. Ha-Fm g whet; Ear 9M rm 75,.4 IlFfaII FIMII a; P M4, WIN, Fltmxtd I 3'6 JFz-I rPFQW é) Lai- Hrmxfiri-WI tug k H =7 F512) Haze: ,3' 94:1 1% en; n? 3m $351.5 Lara. udqvm? ...
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