FinalTestReviewSpring2011

1 1 010 0 101 1 101 0 010 1 101 0 0 0 101 1 2 1 010 0

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Unformatted text preview: responding to this value of ! because _dim N!A " ! 3 ! 1 ! 2_________. !2 3 Those corresponding eigenvectors are: v 1 ! " 0 , v2 ! " 1 3 (any two independent 0 vectors in the null space will do ) The characteristic polynomial must have a factor of __! 2 __________ because _! ! 0 has multiplicity at least two due to two independent eigenvectors for ! ! 0________ . The characteristic polynomial of A is p!! " ! __! 2 !8 ! ! "__________________ (calculate below). 3!! 2 !1 6 4!! !2 !3 !2 1!! det ! 8! 2 ! ! 3 The remaining eigenvalue is _! ! 8______ and the corresponding eigenvector is !1 __v 3 ! " !2 _. (Show your work below) 1 Add the second row to the first below to aid in calculations: !5 2 !1 6 1 !4 !2 # !3 !2 !7 !2 !3 6 !4 !2 !3 !2 !7 1 !2 !3 # 0 1 2 0 0 0 1 !2 # !3 0 16 8 0 !8 !16 101 # 012 000 2 3. (6 pts) !1 101 If I tell you that the eigenvectors of A ! are 0 , 0 101 1 0 can you easily check? What eigenvalues correspond to these eigenvectors? , how 1 010 0 , 1 1 Take each proposed eigenvector and multiply by A : 101 !1 !1 010 0 101 1 101 0 010 1 101 0 0 0 101 1 2 1 010 0 101 1 0 ! 0 !0 0 1 0 ! ! 1 0 2 so ! 1 ! 0 0 0 !1 !2 1 0 so ! 2 ! 1 so ! 3 ! 2 1 What eigenvalues correspond to these eigenvectors? Given that A is symmetric, it should have a complete set of orthonormal eigenvectors? What vectors (explicitly) are those...
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