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Lecture_4_Mathematics_Scanned.pdf - Eigenvalues and...

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o o o a a o Eigenvalues and Eigenvectors Applications in Mathematics Physics Chemistry Biology Statistics Finances
Eigenvectors and Eigenvalues Li near Transformations
Eigenvalues and Eigenvectors Let A be a square 2x2 matrix Ax-b Lets consider the transformation: Ax -+ y v+E Ax T .'1 r" 2 ":[i) u - [;'), -2 0 (z o=[, 1' , It ^t tl Ê ^( 2 .tu -f HüUnË tr ßllbets ttl'nultiplicaticn hv ¡t ,') = [-i i):[i) : )( )[ 0 0 (z v-Au=l " [1 (z v-Au-l " [1 -2 -2 2 1 A 2v v
a Eigenvalues and Eigenvectors : Definition: An eigenvector of nx n matrix A is a nonzero vector x such that for some scalar A scalar 2 is called an ellgenvalue. of matrix A if there is a nontrivial solution x of T* - ),x Such x is called an eigenvector corresponding to 7 Ax .Lx o o
Eigenvalues and Eigenvectors It is easy to determine if a given vector is an eigenvector of the given matrix ) A:I Au Av 1 5 6 2 6 ) u- 6 2 -5 6 2 -2 ": [_") ):e[i *U =2 -5 ):I ):I 11 J )[ )[ (l Il È : -4 ); -24 20 li) ;l) .^(:,) u t1 l r .,1n 3 \ .'1t t l(Ì 3{r =2V Ar/ 3 aV 1'¡ *ltt TT -":{} ..1u .= *Ju. but o1r'+ ¡.r'
Eigenvalues and Eigenvectors Show that 7 is an eigenvalue of mãtrix A. Ax - 7x Ax: ,Lx .=[ (t A:l Is ìK 6 2 Il 6 2 xr xr )(:,)-' 1 0 xr x2 Yt [2 p 9 ¿ 6 ¿ )(:,) '[;) 1 5 1 5 (t [' fx( X ï V¿ I -0 .T il;)=o | '(å i))[;):.
Eigenvalues and Eigenvectors Show that 7 is an eigenvalue of matrix A. Ax-7x Ax-.Lx xr x2 -'[åi) ) ) 6 Il Il x2 x2 :) 0 0 -? Y¿ -1 .-' -6 A, -+ & l(-6) Rr)R215 xr-xz:0 t xl 00 x, is free var x2 T6 x2 7 52 x2 7 ):[:) ):[:) I; [;;) ) 1 1 : Itl : ,lûì. lt ) '[tJ' - 2 5 1 -1 1 1 o'1 o) R, è Rr- R, 5 -5 ",(l) "rl ,r)= X ¿ {1 16 1 x2 x2 v- A Eigenvectors are not unique. ,4v=7V -üJ.r(* X, tP- Xrþ 0
Eigenvalues and Eigenvectors [[r Ð-(:;)l;) [:) ('-,^ ,:^)(:,)--[:) o{ttw + o w ß- h-r 6 \l o't 5 ,- ^)l:o $6 Kx=o lu -C >J¿=-(. I -I,t?=[- (o k -f- (1 - 2)(2- ,L)- 30 : 0 2-),-U"+l-30=0 -37 -28 : 0 characteristic 4:7, 7z: -4 o<p 0 k K equation ß -\ 6
Eigenvalues and Eigenvectors The eigenvalues are 7 and -4 Ax - 7x Ax - To find corresponding eigenvectors ,:À ('-,^ (i) Il -4 -4 x )"2 [;) A 5 1 1 1 0 0 2-L A, + & l(-6) Rr)R215 6 2-7 Rr)Rr-R, 1 l:) 6 [;) =(:',] =",[l) x2 xr )(:,):(i:',): ):[;):'[1) 10 10 t-7 5 1 -1 0 000 [l ;)( xL-xz:0 x, is free var " : [l)' 5 x2 x,2
Eigenvalues and Eigenvectors The eigenvalues are7 and -4 Ax - 7x Ax - To find corresponding eigenvectors 6 2-1 6 -4 -4 x )"2 h:7 5 [' [' x2 ;)[ xL x2 :,) )" :) )[ :-o' , -ur-ot :) [ :[:) ['- 1 -6 ç; l/z 4 ,L2 s 2- 1, -+ 6 0,0 0 0 6 5 s 6 0 5 6 0 R, ) Rr- R, 5q + 6x, -g x, is free var 6 .A' 5 xr 2 V "": I 5 x2 ): ):[-î) : 6 5 x2 x2 -6 1 5 5 ) , ft ¿ VL
Eigenvalues

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