The Eigenvalue Problems - 8.8
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1 The Eigenvalue Problems Definiion of Eigenvalues and Eigenvecors: Le A be an n! n marix. A scalar s said o be an eigenvalue of A if he linear sysem Av!v has a nonzero soluion vecor v. The soluion vecor v is said o be an eigenvecor of A corresponding o he eigenvalue!. The pair!, v is called an eigenpair of A. Noe ha he lef side of he equaion is a marix vecor muliplicaion and he righ side is a scalar vecor muliplicaion.!, u, v, v!. Deermine if each of u, v, v is an eigenvecor of A. If so, wha is is eigenvalue? Skech he graphs of u, v, v and Au, Av, Av. Au Av Av!,souis no an eigenvecor of A. v!v,sov is an eigenvecor of A corresponding o he eigenvalue!. v,sov is an eigenvecor of A corresponding o he eigenvalue!! u,... Au - v,... Av - v,... Av How can we find all pairs of eigenvalues and eigenvecors of a given A? Observe ha an eigenpair saisfies he equaion: Av!v, ha implies ha Av!!v!A!!Iv. Since v, he homogeneous sysem!a!!iv has a nonzero soluion if and only if he marix A!!I is singular or equivalenly de!a!!i. So, if s an eigenvalue of A hen i is a soluion of he equaion de!a!!i. Noe ha de!a!!i is a polynomial of degree n and he equaion de!a!!i has he mos n soluions.. Seps of Finding Eigenvalues and Eigenvecors: () Se he marix A!!I. () Compue P!! de!a!!i. Solve P!! for!. () For each! obained in (), find all nonzero linearly independen soluions of he sysem (A!!Ix.!, v, B! eigenvecors of A, B, and C.! 7, C. Find he eigenvalues and
2 a. Eigenvalues and eigenvecors of A : i. Se A!!I!!!!!! ii. P!! de!a!!i!!!! P!!!!!!!!!!! or!!!!, and! iii.!, solve (A! Ix. A! I!! R R $ R! x! x, x x, x!, solve (A! Ix.!! A! I!!!R R $ R!! -x! x, x!x, x!!! eigenpairs of A :(,,!, b. Eigenvalues and eigenvecors of B : i. Se B!!I!!! 7!! ii. P!! de!b!!i 5!!! P!! 5!!!,! 5,! 5 iii.! 5, solve (A! 5Ix. A! 5I!!! R R $ R!!x x, x x x, x eigenpair of A: 5, c. Eigenvalues and eigenvecors of C : i. Se B!!I!!!!! ii. P!! de!b!!i!!! P!!!! $i! $ i iii.! i, solve (A!! iix. A!! ii!i!!i!ir R $ R!i!
3 !ix! x, x!ix, x!i!i i! eigenpairs of A : i, i!,,!!!!!!!, Find he eigenvalues and eigenvecors of A. a. i. Se A!!I!!!!!!!!!!!! ii. P!! de!a!!i 5! 5!!!! P!! 5! 5!!!!, Soluion is: %! &,%! 5&, %! 5& iii.!, A! I!!!! A! I!!!! Gaussian Eliminaion!! x! x x x ; x! x! x x!x x x x x! 5, A! 5I A! 5I!!!!!!!!!!!!!!! Gaussian Eliminaion!!!!!!! x! x! x x!!x x x!!s s!s s! s!! eigenpairs of A are:
4 ,, 5,!,! Example Le A be a! marix. Find all eigenpairs of A if we know A has eigenpairs i i i,, and,.! i i Rewrie he given wo eigenpairs: i,! i, and, i! Then he oher wo eigenpairs are: i,!, and i,! Example Wha are he eigenvalues of a diagonal marix, and an upper riangular marix? Le D diag%d,...,d n & a diagonal marix and U!u ij, u ij for i % j an upper riangular marix. Observe ha D!!I diag%d!!,...,d n!!& is again a diagonal marix. So de!d!!!d!!...!d n!! and! d,...,d n. Observe also ha U!!I is again an upper riangular marix wih diagonals u!!,...,u nn!!.hence, de!u!!i!u!!...!u nn!! and! u,...,u nn.. TI89: nd Mah, (Marix), eigvc(a) for eigenvecors and eigvi(a) for eigenvalues. Noe ha he eigenvecors are normalized o uni vecors.. Properies: a. A is singular if and only if! is an eigenvalue of A. Proof Le A be singular. Then de!a. Since de!a! I de!a,! isan eigenvalue of A. Now le! be an eigenvalue of A. Then de!a! I. Since de!a de!a! I, A is singular. b. If!!,v is a pair of eigenvalue and eigenvecor of A, hen!!,v is also a pair of eigenvalue and eigenvecor of A for any scalar. Proof Le!!,v be a pair of eigenvalue and eigenvecor of A. Then Av!v. Then
5 A!v!Av!!v!!v. So,!!,v is a pair of eigenvalue and eigenvecor of A. c. If!!, v is a pair of eigenvalue and eigenvecor of A, hen!! k,v is a pair of eigenvalue and eigenvecor of A k. Proof Le!!,v be a pair of eigenvalue and eigenvecor of A. Then Av!v. A k v A k!!av A k!!!v!a k! v...! k! Av! k v. Therefore,!! k,v is an eigenpair of A k. d. If!!,v is a pair of eigenvalue and eigenvecor of a nonsingular A, hen!!,v is a pair of eigenvalue and eigenvecor of A!. Proof Since A is nonsingular, all eigenvalues are nonzero and A! exiss. Le!!,v be a pair of eigenvalue and eigenvecor of A. Since! and A! exiss, Av!v A! Av A!!!v!!A! vv!!a! v! v!a! v Hence!!,v is a pair of eigenvalue and eigenvecor of A!. e. Eigenvalues of A and A T are he same. Proof Observe ha de!a T!!I de!!a T!!I T de!a!!i. So, eigenvalues of A and A T have he same eigenvalues. Noe ha he eigenvecors of A and A T corresponding o! may no be he same. 5
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