LA_Lecture16

LA_Lecture16 - Lecture 16 Eigenvalues and Eigenvectors Last...

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Unformatted text preview: Lecture 16 Eigenvalues and Eigenvectors Last Time-Matrices for Linear Transformations-Transition Matrix and Similarity Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE e-NTUEE SCC_01_2008 16- 2 Lecture 16: Eigenvalues and Eigenvectors Today Eigenvalues and Eigenvectors Diagonalization Symmetric Matrices and Orthogonal Diagonalization Reading Assignment : Chapter 7 7.1 Eigenvalues and Eigenvectors Eigenvalue problem: If A is an n × n matrix, do there exist nonzero vectors x in R n such that A x is a scalar multiple of x Eigenvalue and eigenvector: A a an n × n matrix e e a scalar x a nonzero vector in R n x Ax λ = Eigenvalue Eigenvector Geometrical Interpretation 16- 3 Ex 1: (Verifying eigenvalues and eigenvectors) - = 1 2 A = 1 1 x 1 1 2 1 2 2 1 1 2 x Ax = = = - = Eigenvalue 2 2 ) 1 ( 1 1 1 1 1 2 x Ax- = - = - = - = Eigenvalue Eigenvector Eigenvector = 1 2 x 16- 4 Thm 7.1: (The eigenspace of A corresponding to λ ) If A is an n × n matrix with an eigenvalue λ , then the set of all eigenvectors of λ together with the zero vector is a subspace of R n . This subspace is called the eigenspace of λ . Pf: x 1 and x 2 are eigenvectors corresponding to λ ) , . . ( 2 2 1 1 x Ax x Ax e i λ λ = = ) to ing correspond r eigenvecto an is . . ( ) ( ) ( ) 1 ( 2 1 2 1 2 1 2 1 2 1 λ x x e i x x x x Ax Ax x x A + + = + = + = + λ λ λ ) to ing correspond r eigenvecto an is . . ( ) ( ) ( ) ( ) ( ) 2 ( 1 1 1 1 1 λ λ λ cx e i cx x c Ax c cx A = = = 16- 5 Ex 3: (An example of eigenspaces in the plane) Find the eigenvalues and corresponding eigenspaces of - = 1 1 A - = - = y x y x A 1 1 v If ) , ( v y x = - = - = - 1 1 1 x x x For a vector on the x- axis Eigenvalue 1 1- = λ Sol: 16- 6 For a vector on the y- axis = = - y y y 1 1 1 Eigenvalue 1 2 = λ Geometrically, multiplying a vector (x, y) in R 2 by the matrix A corresponds to a reflection in the y-axis. The eigenspace corresponding to is the x-axis. The eigenspace corresponding to is the y-axis. 1 1- = λ 1 2 = λ 16- 7 Thm 7.2: (Finding eigenvalues and eigenvectors of a matrix A ∈ M n × n ) ) I det( =- A λ (1) An eigenvalue of A is a scalar λ such that ....
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This note was uploaded on 09/12/2011 for the course E 101 taught by Professor J during the Spring '11 term at Adrian College.

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LA_Lecture16 - Lecture 16 Eigenvalues and Eigenvectors Last...

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