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lab6_sol_f_08

lab6_sol_f_08 - Math 115 Lab 1 Fall 2008 Topics 0...

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Unformatted text preview: Math 115 - Lab 1 - Fall 2008. Topics: 0 Eigenvalues and Eigenvectors, o Diagonalization, 0 Linear Dynamical Systems You are to provide full solutions to the following problems. You are allowed, and encouraged to collaborate with your classmates, use your notes and textbook and ask the TA for guidance. Direct copying of solutions is not encouraged, nor is it allowed or ethical. Last name: First name: ‘ Student number: Math 115 - Lab 1 - Fall 2008. Student number: 1 0 3 3 . 0 1 a) Use the characteristic polynomial cA(m) of A to show the eigenvalues of A are 1. Consider the matrix A Now -3,1,and4. ".. -,I «2 o imam x4 x—‘Z *3 WM? x—‘f X—‘f XJf {A001 1’; x -3 3;“ ~3 >< ~3/TL‘; ‘3 X *3 o '2 x»! o —2 x4 0 ~2 x4 C243! x-‘f b x“? C2—cl x—V 0 0 . jLI‘77 Xi? -3/ :3 35 x+3 0/ :(x—9‘)(X+3)(Kw/) 0 ~2 x4 0 ~12 XII 17%2 eyeihi/a/ues 01C ,4 are -3/ // 4/70/71 b) What are the multiplicities of each eigenvalue of A? £6164 eiythIII/ae M5 a Wig/leﬂ/f‘C/‘A/ 07p 0/76, ,, . ~ __, - -2 0 4 I 1’2 a A» 3. x1441}: ’3 4] 454/ [_3 J; #53210 D ,2 J! O ‘2 —'2‘_ o c) Find the set of eigenvectors corresponding to each eigenvalue of ””7“ I 0 I ’ / -. X »X :0 3 0 l 1 I 3" LZIL X ﬂ3+2R2 O 0 o] 3 . , 5 4 0 man '31-? :7“ 'I ’l/S I /’>IH[’3‘,ZZM~?0-1D]3;OI0 0 “1 o JAM 0 ‘2 o A. +X3 7 0 L671 X3 3-6 , 3 [[3611 Vader! cares/0601;??? X1“; 0 V . 710 / are Xi £23" arL‘Tr’W [i3] 4" [ J i ‘fikI’AT ~Z arm 3 —2 773:1?! [I J/z 7717/ 9/; 0] ~3tf—3ﬂpiam o 2_3....?0 ( 3/2 d) Give an invertible matrix P and a diagonal matrix D such that P‘lAP = D. ,IrH' ,,£3007 Pfij-i o 3/2] D" 010 Igloo 72x22 /—11100 0 “( 0 I M 0 I o '99.. 0 V2 37/7,! 1 I 1 1412‘? o 0 { 1/? 1/? 9/? 9 1/2 c9 ALEX/~03 / 0 o (if 47/? %¥] 0 “72 0 V7. W9 0 I 0 “’1 o ’/2 l 1/9 2/? 1/? 0 o 1‘ 2/? Q7; 1/? f) Using the equation P 1AP: D, give the matrix A2. “L: 1,}:)_’ 0,00 A pr (Z 03/7]o!0]/1[?0?]: (I 0016]” 11674143933 ”ﬁn/82:12"! '7' 0? 0] 3 [6 V ‘f 9‘ g) Given the linear dynamical system Vo— — [—2] and sz AkVo 3 2 —1 —2 . . 0 1 2 1 2. Conmder the matrix B — 0 0 1 _1 . 0 0 0 2 8.) Use the characteristic polynomial cB(x) of B to show the eigenvalues of B are 1,2,3. CgKﬂ‘" X”? ”Z / Z :(anffplﬂx-Z) I ° o X’{ I ~. L" W 6: , EyéﬂVmueﬁ 4 b) What are the multiplicities of each eigenvalue of B? / 5515 /77L{/71/;0//c{{y 41,410, 2 and! 3 AQVQ ma/vz/p/I'C/‘I/y 0/?6'. 42, am/ 3 c) Find the set of eigenvectors corresponding to the eigenvalue 1 of B. t: , 3 -2 4 l 2 ﬁle? I ,_t/ _/ l (1 0 3/? >\ [ AI B 0 o -1 ’1 7/122 3:: :7" [/1 WW“ :; c: V; o; If t) D D 0 Iii/+2133 M/ l b 0‘ m3“? 0910 ,,)(!+X2*0 It {’6 ﬂz—zm 33;; K530 Le, 2’ Xry’i’D t . , . X »/ ,3 Tim, e/je/Wcho/P; Cohl’ESﬂohJM] 110- / are, x: - f ( r3 " 0 J )0! D (1) Explain whyB is diagonalizable or not. 100 r. dyer/T‘Arl/ Eijcha/Me X3/ ha? Mu/if‘p/I/a'fy fit/Va half the yanle/ 50/145”; 079 AZ’Qro Ag; Dn/y DMZ para/wafer- 5 LB [5 NOT d/‘ayoﬂﬂ/L‘Zaé/e. 3. If A = PP‘1 for some invertible n X 71 matrix P, give the eigenvalues for A With their multiplicities. H2/0/0'I7PI/a'l “3/07?sz / is Hm, eigenvﬁ/ae (:27!) x4 «tr/2% MQI/%W\$CI‘/P W, 4. If A is an eigenvalue of A, give an eigenvalue for the matrix A2 + 5A — 4I. LQ‘IL X be a /\—€f_92b!/€C‘?lor“ D70 f7 1 5,091+9ﬂv1/KJX: HZXFI— 979x”#’l>(:)‘ X + S .. (mew/M 4'. Xltf/k’q )5 4h e/‘yEﬂV’Q/qél 07(‘ ll] )X’¥X 1% 9!? ~ ‘ff. 5. If A2 = AA, how are the columns of A related to the eigenvectors of A? Explain. LQYL I4; 49 #22 Ni" column 07¢ ﬂ- 41:53.41" ,:/ii-/);:)\/4;} /fi 14.40. 'ﬁn]:>\[ﬂlﬂhj ,I/ 7/18 [Dali/ﬁn“; 07" ﬂ 4P9, elje/erc7ZO/‘5 076‘ ﬂ. 6. Bonus Question: Give the basic eigenvectors of all diagonal matrices. If D3615] g; a aljajoriai mafr-ﬂixj if Ms 4+ mow/- gnizloy in The, éaérc eigenveciow of ali gl/ajo/y 0&6 Hon " gem: amt/i r‘DW, ”17C X (If; a Vacil’ﬂr‘ net/[Ht oniy one hmw“Z‘?—V‘0 {unity in [5'59"{5‘091 I.) ’HIZIJ szd;;X cz/ maﬁ/‘c‘e; WY? a// Wafers MM: (SCQCHV me, nan/ZEN? @nVZk/y, ...
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lab6_sol_f_08 - Math 115 Lab 1 Fall 2008 Topics 0...

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