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# Class Note for MATH 3331 with Professor He at UH

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This 29 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

x h at max I 4 4 139 Lecture 23 9 cmmz cmsz Wm 5mm 9 0 wa MTzchmqu 39r mv Milky Wm H Wm M WM WW M mm m MawwmZwm Homogeneous System Homogeneous system QN FHR 39 X AX J A constant n x n matrix 39If n 1 513 2 cm gt 2375 2 C39th JAM 4quot D 7quot 7 n 1 Try exponential form for 1quot A nggkv R quot9 xt ZEYXQ l constant vector 39 vetr Sub Xt in 1 gt EH M 2 9 x 15 33213 axe 1224 M A3 1y Z AVZAV A 3 quotLJ39H Math 3331 Section 19470 Lecture 28 April 13 2009 2 24 Section 91 Overview Eienvalues Eigenvalues and Eigenvectors dr04flh Def A number A is an f eigenvalue of A if there is a vector v 739290 such that Ay 2 AV 2 If A is an eigenvalue then any Q4 v 0 satisfying 2 is called an eigenvector for A u a 1 f U 57 quot00244 39 39 g Ivl iii Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 3 24 Overview Eienvalues Section 91 Solution of Homogeneous System Homogeneous system X AX 1 A constant n x n matrix Thm If A is eigenvalue of A and v is eigenvector for A then xt eVgis a solution of 1 I Mrpmh l 74 April 13 2009 4 24 Math 3331 Section 19470 Lecture 28 Jiwen He University of Houston Section 91 7 Overview Eienvalues Characteristic Polynomials Z w 1 1 s sic s l w z V g a V gt4 AEUa if 7 adv I O id h39l f quotIn using vIvle I a 0quot 339 A AIlVZO tth Def A number is an Since v 72 0 gt detlA AI O eigenvalue of A if there is a w vector v 7k 0 such that AV 2 AV 2 D6f If A is an eigenvalue then any 2 characteristic polynomial v 9 satisfying 2 is called p of 665Wamp an eigzect 0r c 0 Note he degree of 190 is n 7 39 h gt p has n roots a if counted with multiplicities I 139 71 s 7 3 91 Y r f H Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 5 24 9 01 2 IIAJJ39Z as 4 z gt quot4 V d rm 7quot 47 4 SruIvk S va W f d o 2439 wyC L9 Id zi a VJ dl l a 0quot r1 gt Pa 0Mo ow1 amp I O a r a 4 z 39139 0 dArdfxy gt anyIda Section 91 l Overview Ei39envau Finding Eigenvalues and Eigehvectors Thm The eigenvalues of A are the roots of pA detA AI o 3 If A is a root of 3 then any v 7E 0 in hullA AI is an eigenvector for A Def If A isan eigenvalue of A then nullA AI is called the eigehspace of A 11 Math 3331 Section 19470 Lecture 28 April 13 2009 6 24 Section 91 4 O rview Ei envalua Distinct Eigenvalues and Independent Eigenvectors Ci quotquot39u r r Z I f7 gt V Cl 1 it ll It 02 Thm ELgenvectors for distinct eigenvalues are linearly independent Ap kfskhvu AU 23 c z uz39 u 2 U f a 20 m all 0 S f Clirf Jtquot g 0111 11 a 2 EC A 6017 Qi Mg 4 2kg Math 3331 Section 19470 Lecture 28 April 13 2009 7 24 Section 91 Fundamental Stl quot a a Consequence If p has n distinctl39eal loots L J A N A1 7 An then A has n linearly indegen dent eigenvectors V1Vn 237 Z lef gt eA1tV e Y2 is fundamental set of solutions Iii A Math 3331 Sectian 19470 Lecture 28 April 13 2009 8 24 a 3sz mama 9 533 a Exercise 915 AC j otE A al c L c d 40064 Z4331 Ex 915 Find pm and eigenvalues by hand for A 2 3 Mg 4 7 A a A chi A S 395 pm A mt 21 9 Trj w r 0 2 DA r MA5 i i x m rix wi gt 2 2 2 1 2 21 Tu W A lt X gt s l b AEcd39 Ogt A 39 pm a39 aquot c M M Jan iA r my I 4 v 40 Math 3331 Section 19470 Lecture 28 April 13 2009 10 24 1 RAJ 0 fa MfAA If J a A wearW7 1 Cu z h 391 A quotquot obiA k J 39 In Class Exercises Exercise 9111 1 4 2 Ex 9111 Find poi and eigenvalues by hand for A p 1 1 6 12 2 x w greys 396 R S j J uquot l L i 0quot 9 egg 5 detA I 3 Z MA 22 1 2 1 4 1gtlt122i 62 gti1 1 6P 12i 1 A 1 2 12 1 12 24 g1A A2igd L41121 gt 1 A 2 lt AgtltA1gtltA 2 gt Eigenvalues 112 In Jiwen He University of Houston Math 3331 Sectiqn 19470 Lecture 28 April 13 2009 ll 24 39 In Class Exercises T mS Qe 439 39Ex 9127 Find fundamental solution set by hand for y Ay if 3 2 A 6 3 12 2 2 6 ii is 1quot 1 3 Q 2 399 1 p detA I 6 3 A 12 2 2 6 11 3 12 13 6 3 lt b3A 2 6A lt lug2 2 3 A3 A 6 A 24 212 23 A 43AXA23A6gt 4A3 A3A23A2 A 3A 1A 2 gt eigenvalues 51 1gt2 2 A3 3 Find eigenvectors H V31 I 143 Jt Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 12 24 In Class Exercises Exercise 9127 cont 1 A1 1 2 O 2 l O 1 AI 6 4 12 R31gt12 6 4 12 2 2 2 2 5 1 Q 1 1 O l R12167Rgt13 1a 2 O 4 6 A 0 2 3 O 2 3 0 o 0 Set free variable y3 2 gt y2 3y1 2 i eigenvector v1 2321 A H 2 A2 2I 1 0 2 1 0 2 1 o 2 A2I 6 5 12 R12 1 13 1 2 0 5 0 gt 0 1 0 2 2 4 0 2 0 0 o 0 0 Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 13 24 In Class Exercises Exercise 9127 cont 3 A3 3 O O 2 1 1 O A 3 6 6 12 gt O O l 2 2 3 O O 0 Set free variable y2 1 gt 33 0311 1 gt eigenvector V3 2 11OT gt fundamental solution set 2 2 Y1t 715 3 7 5720 6727 0 7 Y3t 67316 2 1 Ol l l 26439 26 e 31 Note Associated fundamental matrix is Yt 3e t O 2 316 26 e 2t 0 General SOIUUOW yt 01Y1t C2Y2Ct 03Y3t YUM C 01 C27 031T F1 Jiwen He University of Houston Math 3331 e Section 19470 Lecture 28 April 13 2009 14 24 In Class Exercises Exercise 9136 Ex 9136 Find eigenvalues and eigenvectors using a computer for 6 5 9 10 W f 10 7 13 16 A 4 4 8 8 639 5 3 5 7 1 Numerical computation via Matlab s poly roots and null commands gtgt A6 5 9 101O 7 13 164 4 8 8 5 8 5 7cpolpolyA cpol 10000 20000 10000 20000 00000 The output of poly is a row vector whose entries are approximated values for the coefficients of the characteristic polynomial pm x 10000 x A4 20000 x A3 10000 x A2 20000 x A 00000 Iiii Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 15 24 In Class Exercises Exercise 9136 cont Find the roots of the characteristic polynomial gtgt evalsrootscpol evals 10000 20000 ruJ I AAfb 10000 00000 So the eigenvalues roots ofp are approximately 10000 20000 10000 0000 They can be accessed via evas1 evas2 etc Now compute bases for the ullspaces of the eigenvalues using the null command 2 I a 639 gtgt v1null evals1eye div 4 V1 k J Ijgt 05774 05774 w 00000 05774 The n x n identity matrix is denoted in Matlab by eyen here n 4 I Analogously one can compute the other three eigenvectors Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 16 24 In Class Exercises Exercise 9136 cont 2 Symbolic computation using Matlab s poly factor or solve and null commands poly and null work also for symbolically defined matrices The roots command works only for numerically defined vectors To find roots of a symbolically defined polynomial use the commands factor or solve gtgt symAsymAsymcpolpolysymA symcpol 1 4 2x3 x 22x 4 In 0 c Luna D g Note that here the output is a symbolic polyn mial expression with default variable v Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 17 24 In Class Exercises Exercise 9136 cont You can find the eigenvalues with the factor command gtgt factorsymcpol ans XX 1X 2X1 So the exact eigenvalues are A1 0 A2 1 A3 2 A4 1 Alternatively you can find them using solve gtgt symevalssolvesymcpol symevals O 1 E 2 1 Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 18 24 In Class Exercises Exercise 9136 cont Now find eigenvectors gtgt symv1nu11symAsymevals1 eye 4 symv1 E 1 E 1 E 1 E 1 hence v1 1111T Analogoust one finds the eigenvectors for A2 A3A4 V2 2 01 27071T1 V3 E1107271TV4 19 1907 Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 In Class Exercises Exercise 9129 a avslr quot Ex 9129 Find eigenvalueQrRl eigenvectors using a computer for Eigenvalues and eigenvectors can be computed directly in Matlab with the eig command Outputs 7 2 10 A 0 1 0 V matrix whose columns are eigenvectors 5 2 8 D diagonal matrix whose diagonal entries are eigenvalues April 13 2009 20 24 Math 3331 Section 19470 Lecture 28 Jiwen He University of Houston 39 quot ln Class Exercises Exercise 9129 cont WithGut SD CifiCEltiOn outputs are Symbolic computation yields exact flOatlng DOlnt numbers values if available A7 2 10O 1 O5 2 8 A7 2 1oo 1 o 5 2 8 VD eigA VD eigsymA V V O8944 O7071 O5774 1 2 1 O 0 05774 1 O O O4472 O7071 O5774 1 1 1 D D 2 O O 1 O o 3 o O 2 O o o 1 0 O 3 1 2 1 Hence A121V1 1 A22 2V2 O A323V3 O l 1 l Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 21 39 In Class Exercises Exercise 9139 39Ex 9139 Find fundamental solution set via computer for y Ay if 20 34 10 A 12 21 5 2 4 2 Editing A in Matlab and applying Matlab s eig command to symA yields the following eigenvalues and eigenvectors 1 4 v1 1 11T A2 2 V2 211T A3 3 V3 2 210T gt fundamental solution set y1t e 4tl 17 1711T y2ltt e 2tl2711T y3t e3ii27 101T Aril 13 2009 22 24 Math 3331 Section 19470 Lecture 28 Jiwen He University of Houston In Class Exercises Exercise 9149a 6 8 EX 9149i Find determinant and eigenvalues of A 4 6 via computer Describe any relationship between eigenvalues and determinant No computer necessary to find detA 4 Eigenvalues using Matlab A1 27 A2 2 hence MM 2 4 detA Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 23 24 In Class Exercises Exercise 9151a Ex 9151i Find eigenvalues of A a 2 via computer Describe any relationship between eigenvalues and triangular structure of A Matlab gt eigenvalues A1 2 A2 4 These are the diagonal entries of A Thm The eigenvalues of a lower or upper triangular matrix are the diagonal entries Jiwen He University of Houston Math 3331 Section 19470 Lecture 28 April 13 2009 24 24

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