LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.
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1 LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v () We might also write v as v Both notations refer to a vector (2) A vector can be man different things depending on the contet A vector might be a row matri or even a function Here, a column matri will suffice Given a matri A, there alwas eists a number λ and a collection of non-zero vectors v λ such that A v λ λ v λ This means that multipling v λ b A doesn t reall change the essence of v λ Definition We sa λ is an eigenvalue and v λ is an eigenvector associated to λ Note 4 In German, eigen means something like same So this is sort of like saing that this vector isn t essentiall changed b multipling b A Remark 5 The zero vector (ie, the column vectors with all zeros denoted or ) does not count as an eigenvector because A for all matrices A and so it doesn t give ou an information about the matri itself 2 E 6 Let A Then Eigenvalue One Associated Eigenvector λ 2 v λ 2 v 2 But there are man more eigenvectors for each eigenvalue In fact, all eigenvalues associated to λ 2 are of the form t w
2 2 MATH 62 for some t and all eigenvectors associated to λ 2 are of the form w 2 t for some t Eercise Check that and Now, the question is: how do ou find eigenvalues and eigenvectors? a b Let A and let I (I is the 2 2 identit matri) c d Fact 7 If λ is an eigenvalue of A, then λi A is a singular matri Recall that a singular matri is a matri whose determinant is zero So, to find the eigenvalues of A, we need to find the λ such that det(λi A) First, let s figure out what λi A is We write: a b λi A λ c d λ a b λ c d λ a b c λ d Then det(λi A) λ a b c λ d (λ a)(λ d) ( b)( c) (λ a)(λ d) bc This means we need to find the λ such that Eample Let A 2 (λ a)(λ d) bc Find the eigenvalues of A
3 MATH 62 Solution: If λ is an eigenvalue of A, then det(λi A) So, det(λi A) λ 2 λ (λ )(λ) ()( 2) λ 2 λ + 2 Therefore, our eigenvalues are λ, 2 (λ )(λ 2) After we have found an eigenvalue, we want to find the associated eigenvectors Eample 2 Let A be as in the eample above Find an eigenvector for each eigenvalue of A Solution: We know that the eigenvalues of A are λ, 2 We find the associated eigenvectors for λ first and then λ 2 λ : Assume that is an eigenvector associated to λ Then, b the definition of an eigenvector, we must have λ 2 We want to find a relation on and that will make this true So, we multipl the matrices on the left: () + ( )() 2 2() + () 2 B the equation above, we must have 2 This gives use two equations: 2 We see that both of these equations give the same information: 2 So, 2 describes the eigenvectors associated to λ For a specific eigenvector associated to λ, we can take to be an number Sa Then 2 is an eigenvector associated to λ
4 4 MATH 62 Eercise 2 Check that Note 8 If ou had written and then put 2 2, is also true Both 2 and 2 describe the collection of eigenvectors associated to λ λ 2: Let be an eigenvector associated to λ 2 Then the following must be true: 2 2 Using matri multiplication on the left, we get This again gives us two equations: B the second equation, we see that Hence, the eigenvectors associated to λ 2 look like or For a specific eigenvector, we might take Thus is an eigenvector associated to λ 2 Eercise Check that 2 2 Eample Let A Find the eigenvalues of A and for each eigenvalue find an associated eigenvector Solution: We alwas start b finding the eigenvalues of A
5 MATH 62 5 Eigenvalues: Write det(λi A) λ λ λ λ (λ )(λ ) ( )() λ 2 4λ + + λ 2 4λ + 4 (λ 2) 2 λ 2 is a repeated eigenvalue, which is fine This actuall makes our work easier because now we onl need to find the eigenvectors for one eigenvalue Eigenvectors: Let be an eigenvector for λ 2 Then Using matri multiplication on the left, we get So our equations are: From this, we get that Hence, eigenvectors associated to λ 2 are of the form Eample 4 Check if either v 5 or w are eigenvectors of A If so, find the associated eigenvalue Solution: The eas wa to do this is to simpl multipl A b v and w and see what happens
6 6 MATH 62 Let s start with A v We write 2 5 A v 4 2( ) + 5() 4( ) + ( )() ( ) 7() 7 7 v Therefore, v is an eigenvector associated to λ 7 Net, we check w Write A w 4 2(5) + 5(5) 4(5) + ( )() Now, the question is: does there eists a λ such that 5 5 5λ λ λ If there did, we would get 5 5λ λ But the first equation implies λ and the second implies λ So no such λ eists Therefore, w is not an eigenvector of A Eercise 4 Show that λ 2 is the other eigenvalue of A Eample 5 Find the eigenvalues and eigenvectors of 2 5 A 2 Solution: We start with the eigenvalues
7 MATH 62 7 Eigenvalues: Since we write λi A λ 2 5 λ + 2 det(λi A) λ 2 5 λ + 2, (λ 2)(λ + 2) ( )( 5) λ λ 2 9 (λ )(λ + ) Therefore, our eigenvalues are λ, Eigenvectors: Because we have two eigenvalues, we need to break this into cases: λ : Let be an eigenvector associated to λ Then λ Using matri multiplication on the left and scalar multiplication on the right, we get The equations we get from this are: Solving, we find Hence, the eigenvectors associated to λ are of the form A specific eigenvector can be found b taking : λ : Let be an eigenvector associated to λ Then we must have Simplifing on the left and right, this equation becomes
8 8 MATH 62 So we get the equations Hence, 5 So eigenvectors associated to λ are of the form 5 5 A specific eigenvector can be obtained b taking :
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