Lecture_20 - Lecture Note 20 Dr Jeff Chak-Fu WONG...

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Lecture Note 20 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1
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E IGNEVALUES , E IGENVECTORS , AND D IAGONALIZATION 1. Eignevalues and Eigenvectors 2. Diagonalization 3. Diagonalization of Symmetric Matrices E IGNEVALUES , E IGENVECTORS , AND D IAGONALIZATION 2
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D IAGONALIZATION OF S YMMETRIC M ATRICES D IAGONALIZATION OF S YMMETRIC M ATRICES 3
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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. D IAGONALIZATION OF S YMMETRIC M ATRICES 4
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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 y -axes 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. D IAGONALIZATION OF S YMMETRIC M ATRICES 5
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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) D IAGONALIZATION OF S YMMETRIC M ATRICES 6
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Theorem 0.2 If A is a symmetric matrix, then eigenvectors that are associated with distinct eigenvalues of A are orthogonal. Solution Exercise. D IAGONALIZATION OF S YMMETRIC M ATRICES 7
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Example 1 Given the symmetric matrix A = 0 0 - 2 0 - 2 0 - 2 0 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 = 0 , j = 1 , 2 , 3 and obtain the respective eigenvectors (verify) x 1 = 0 1 0 , x 2 = - 1 0 2 , x 3 = 2 0 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). D IAGONALIZATION OF S YMMETRIC M ATRICES 8
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Thus A is diagonalizable and is similar to D = - 2 0 0 0 4 0 0 0 - 1 . D IAGONALIZATION OF S YMMETRIC M ATRICES 9
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We call that if A can be diagonalized, then there exists a nonsingular matrix P such that P - 1 AP diagonal. Moreover, the columns of P are eigenvectors of A .
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