# 6.1 - Math 2940 Solutions Fall 2011 Section 6 1 5 det 3 Î 1...

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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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6.1 - Math 2940 Solutions Fall 2011 Section 6 1 5 det 3 Î 1...

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