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Unformatted text preview: Lecture Note 20 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff ChakFu WONG 1 E IGNEVALUES , E IGENVECTORS , AND D IAGONALIZATION 1. Eignevalues and Eigenvectors 2. Diagonalization 3. Diagonalization of Symmetric Matrices EIGNEVALUES, EIGENVECTORS, AND DIAGONALIZATION 2 D IAGONALIZATION OF S YMMETRIC M ATRICES DIAGONALIZATION OF SYMMETRIC MATRICES 3 In this part we consider the diagonalization of symmetric matrices (an n × n matrix A with real entries that A = A T ). We restrict our attention to this case because symmetric matrices arise in many applied problems. DIAGONALIZATION OF SYMMETRIC MATRICES 4 As an example of such a problem, consider the task of identifying the conic represented by the equation 2 x 2 + 2 xy + 2 y 2 = 9 , which can be written in matrix form as h x y i 2 1 1 2 x y = 9 Observe that the matrix used here is a symmetric matrix. We shall merely remark here that the solution calls for the determination of the eigenvalues and eigenvectors of the matrix 2 1 1 2 . The x and yaxes are then rotated to a new set of axes, which lie along the eigenvectors of the matrix. In the new set of axes, the given conic can be identified readily. DIAGONALIZATION OF SYMMETRIC MATRICES 5 Theorem 0.1 All the roots of the characteristic polynomial of a symmetric matrix are real numbers. We omit the proof of the following important theorem (see D. R. Hill, Experiments in Computational Matrix Algebra, New York: Random House, 1988) DIAGONALIZATION OF SYMMETRIC MATRICES 6 Theorem 0.2 If A is a symmetric matrix, then eigenvectors that are associated with distinct eigenvalues of A are orthogonal. Solution Exercise. DIAGONALIZATION OF SYMMETRIC MATRICES 7 Example 1 Given the symmetric matrix A =  2 2 2 3 , we find that the characteristic polynomial of A is (verify) f ( λ ) = ( λ + 2)( λ 4)( λ + 1) , so the eigenvalues of A are λ 1 = 2 , λ 2 = 4 , λ 3 = 1 . Then we can find the associated eigenvectors by solving the homogeneous system ( λ j I 3 A ) x = ,j = 1 , 2 , 3 and obtain the respective eigenvectors (verify) x 1 = 1 , x 2 =  1 2 , x 3 = 2 1 . It is easy to check that { x 1 , x 2 , x 3 } is an orthogonal set of vectors in R 3 (and is thus linearly independent by Theorem 0.1, c.f. Lecture Note 16). DIAGONALIZATION OF SYMMETRIC MATRICES 8 Thus A is diagonalizable and is similar to D =  2 4 1 . DIAGONALIZATION OF SYMMETRIC MATRICES 9 • We call that if A can be diagonalized, then there exists a nonsingular matrix P such that P 1 AP diagonal....
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This note was uploaded on 05/18/2010 for the course MATHEMATIC MAT2310 taught by Professor Dr.jeffchakfuwong during the Spring '06 term at CUHK.
 Spring '06
 Dr.JeffChakFuWONG
 Linear Algebra, Algebra, Vectors

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