This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture Note 19 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 E IGNEVALUES AND E IGENVECTORS 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 13 (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 EIGNEVALUES AND EIGENVECTORS 4 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 lecture 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. 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 = always satisfies Equation (1), but 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. EIGNEVALUES AND EIGENVECTORS 6 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 . EIGNEVALUES AND EIGENVECTORS 7 Example 2 Let A = 1 2 1 2 . Then A 1 1 = 1 2 1 2 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 = 1 2 1 2 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 ....
View
Full Document
 Spring '06
 Dr.JeffChakFuWONG
 Linear Algebra, Algebra, Vectors, Matrices, Diagonal matrix, Matrix Transformation and Linear Transformation

Click to edit the document details