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Lecture_10_1 - Lecture Note 10-1 Nov 14 Dr Jeff Chak-Fu...

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Lecture Note 10-1: Nov 14 - Nov 17, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 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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E IGNEVALUES AND E IGENVECTORS E IGNEVALUES AND E IGENVECTORS 3
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In this lecture, every matrix considered is a square matrix . Let A be an n × n matrix. Then, as we have seen in Lecture note 3-1 (Matrix Transformation and Linear Transformation), the function L : R n R n defined by L ( x ) = Ax , for x in R n , a linear transformation. A question of considerable importance in great many applied problems is the determination of vectors x , if they are any, such x and A x are parallel (see Figure 1). x x A x A x O O Figure 1: x is an eigenvector of A . A x is in same or opposite direction as x , if λ 6 = 0 E IGNEVALUES AND E IGENVECTORS 4
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Such questions arise in all applications involving vibrations; they arise in aerodynamics, elasticity, nuclear physics, mechanics, chemical engineering, biology, differential equations, and others. In this section we shall formulate this problem precisely; we also define some pertinent terminology. In the next section we solve this problem for symmetric matrices and briefly discuss the situation in general case. E IGNEVALUES AND E IGENVECTORS 5
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Definition: Let A be an n × n matrix. The number λ is called an eigenvalue of A if there exists a nonzero vector x in R n such that A x = λ x . (1) Every nonzero vector x satisfying Equation (1) is called an eigenvector of A associated with the eigenvalue λ . The word eigenvalue is a hybrid one ( eigen in German means “proper"). Eigenvalues are also called proper values , characteristic values and latent values ; and eigenvectors are also called proper vectors , and so on, accordingly. Note that x = 0 always satisfies Equation (1), but 0 is not an eigenvector, since we insist that an eigenvector be a nonzero vector. Remark: In the preceding definition, the number λ can be real or complex and the vector x can have real or complex components. E IGNEVALUES AND E IGENVECTORS 6
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Example 1 If A is the identity matrix I n , then the only eigenvalue is λ = 1 ; every nonzero vector in R n is an eigenvector of A associated with the eigenvalue λ = 1 : I n x = 1 x . E IGNEVALUES AND E IGENVECTORS 7
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Example 2 Let A = 0 1 2 1 2 0 . Then A 1 1 = 0 1 2 1 2 0 1 1 = 1 2 1 2 = 1 2 1 1 so that x 1 = 1 1 is an eigenvector of A associated with the eigenvalue λ 1 = 1 2 . Also A 1 - 1 = 0 1 2 1 2 0 1 - 1 = - 1 2 1 2 = - 1 2 1 - 1 so that x 2 = 1 - 1 is an eigenvector of A associated with the eigenvalue λ 2 = - 1 2 .
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