3.1 Eigenvalues and eigenvectors - Eigenvalues and...

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Eigenvalues and eigenvectors . Definition. Let A = [ a ij ] be an n×n matrix. The number λ is called eigenvalue of A if there exists a nonzero vector x R n such that Ax = λx ( 1 ) . Every nonzero vector x satisfying (1) is called an eigenvector of A associated with the eigenvalue λ. Eigenvalues are also called proper values, characteristic values and latent values; and eigenvectors are also called proper vectors, characteristic vectors and latent vectors. Example 1. 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 eigenvector of A associated with the eigenvalue λ 1 = 1 2 . Example 2. Let A = [ 1 1 2 4 ] . We wish to find the eigenvalues of A and their associated eigenvectors. Thus we wish to find all real numbers λ and all nonzero vectors x = [ x 1 x 2 ] satisfying (1), that is
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[ 1 1 2 4 ] [ x 1 x 2 ] = λ [ x 1 x 2 ] ( 2 ) Equation (2) becomes { x 1 + x 2 = λ x 1 2 x 1 + 4 x 2 = λ x 2 or { ( λ 1 ) x 1 x 2 = 0 2 x 1 + ( λ 4 ) x 2 = 0 ( 3 ) .
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