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Unformatted text preview: Math 2940 Solutions, Fall 2011 Section 6 . 1 5 ) det 3 Î» 1 1 Î» = (3 Î» )(1 Î» ) has roots Î» = 1 , 3. det 1 Î» 1 3 Î» = (1 Î» )(3 Î» ) has roots Î» = 1 , 3. det 4 Î» 1 1 4 Î» = (4 Î» ) 2 1 = Î» 2 8 Î» + 15 = ( Î» 5)( Î» 3) has roots Î» = 5 , 3. The eigenvalues of a sum of matrices are not the sum of the eigenvalues of the matrices. Actually it doesnâ€™t even make sense  there are four possible sums of the eigenvalues of A and B but only two eigenvalues for A + B . 10 ) det . 6 Î» . 2 . 4 . 8 Î» = Î» 2 1 . 4 Î» + . 4 = ( Î» 1)( Î» . 4) has roots Î» = 1 , . 4. An eigenvector for Î» = . 4 is (1 , 1) and an eigenvector for Î» = 1 is (1 , 2). det (1 / 3) Î» 1 / 3 2 / 3 (2 / 3) Î» = Î» 2 Î» = Î» ( Î» 1) has roots Î» = 0 , 1. An eigenvector for Î» = 0 is (1 , 1) and an eigenvector for Î» = 1 is (1 , 2). So A 100 has eigenvalues 1 and ( . 4) 100 , which is close to 0. So A 100 and A âˆž have very close eigenvalues (and it is not a repeated eigenvalue) so they should have very close eigenvectors.eigenvalues (and it is not a repeated eigenvalue) so they should have very close eigenvectors....
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 '05
 HUI
 Linear Algebra, Algebra, Matrices, Eigenvalue, eigenvector and eigenspace, Î», zero radians

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