LA_Lecture16 - Lecture 16 Eigenvalues and Eigenvectors Last...

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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
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16- 2 Lecture 16: Eigenvalues and Eigenvectors Today Eigenvalues and Eigenvectors Diagonalization Symmetric Matrices and Orthogonal Diagonalization Reading Assignment : Chapter 7
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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
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Ex 1: (Verifying eigenvalues and eigenvectors) - = 1 0 0 2 A = 0 1 1 x 1 1 2 0 1 2 0 2 0 1 1 0 0 2 x Ax = = = - = Eigenvalue 2 2 ) 1 ( 1 0 1 1 0 1 0 1 0 0 2 x Ax - = - = - = - = Eigenvalue Eigenvector Eigenvector = 1 0 2 x 16- 4
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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
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Ex 3: (An example of eigenspaces in the plane) Find the eigenvalues and corresponding eigenspaces of - = 1 0 0 1 A - = - = y x y x A 1 0 0 1 v If ) , ( v y x = - = - = - 0 1 0 0 1 0 0 1 x x x For a vector on the x- axis Eigenvalue 1 1 - = λ Sol: 16- 6
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For a vector on the y- axis = = - y y y 0 1 0 0 1 0 0 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
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Thm 7.2: (Finding eigenvalues and eigenvectors of a matrix A M n × n ) 0 ) I det( = - A λ (1) An eigenvalue of A is a scalar λ such that . (2) The eigenvectors of A corresponding to λ are the nonzero solutions of . Characteristic polynomial of A M n × n : 0 1 1 1 ) I ( ) I det( c c c A A n n n + + + + = - = - - - λ λ λ λ λ Characteristic equation of A : 0 ) I det( = - A λ 0 ) I ( = - x A λ Let A is an n × n matrix.
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