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hw5-solutions

hw5-solutions - Assignment 5 Eigenvalues and Eigenvectors...

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Assignment 5: Eigenvalues and Eigenvectors Written by: Saber Mirzaei Problem 1. (a) Solving equation | | | | and (b) Solving equation | | | | and Problem 2. (a) We know that is an eigenvalue of a matrix iff such that ( ) . Also from properties of determinant of a matrix we know that ( ) ( ) , so: So if ( ) then (( ) ) . From properties of transpose of a matrix we have ( ) ( ) . Hence: is an eigenvalue of a matrix iff such that ( ) . Accordingly based on the definition of eignetvalue, is also an eigenvalue of as well (Just notice that all the equalities are two sided so no need to prove the other side for iff). (b) Let be a vector in whose entries are all ’s. Doing the math we get [ ] . Hence is and eigenvalue and is an eigenvector. (c) Using previous part we have is and eigenvalue of . Using part (a) we have is and eigenvalue of as well.
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Problem 3. (a) Assume is the corresponding matrix of transformation . Based on the definition of we have for all nonzero vectors on the line we have ( ) which means . Hence is one of eigenvalues of .
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