ME581_Assignment04-Sol

ME581_Assignment04-Sol - ME 581 Assignment #4 Eigenvalue...

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1 ME 581 Assignment #4 – Eigenvalue problem In the following questions show the derivations that you used to write your programs. 1. For the following matrices: ( do manual calculations ) a. = 3 0 0 1 2 0 4 2 1 ] [ A b. = 1 1 4 1 ] [ B Calculate the determinant. Calculate the eigenvalues. Calculate the eigenvectors. a. Since = 3 0 0 1 2 0 4 2 1 ] [ A is a triangular matrix, therefore the determinant is the product of the diagonal terms 3 2 1 ] det[ × × = A 6 ] det[ = A 0 3 0 0 1 2 0 4 2 1 ] det[ = = λ I A ( ) 0 ) 3 )( 2 )( 1 ( ) 0 ( 4 ) 0 ( 2 0 1 ) 3 )( 2 ( ) 1 ( = = × Eigenvalues of ] [ A In general the eigenvalues of a triangular matrix are the diagonal terms. Eigenvectors: = = ) 1 ( 3 ) 1 ( 2 ) 1 ( 1 ) 1 ( 3 ) 1 ( 2 ) 1 ( 1 1 2 0 0 1 1 0 4 2 0 1 3 0 0 1 1 2 0 4 2 1 1 } ]{ [ X X X X X X X I A We assume 1 ) 1 ( 1 = X 0 4 2 ) 1 ( 3 ) 1 ( 2 = X X and 0 ) 1 ( 3 ) 1 ( 2 = + X X 0 ) 1 ( 3 ) 1 ( 2 = = X X = = ) 2 ( 3 ) 2 ( 2 ) 2 ( 1 ) 2 ( 3 ) 2 ( 2 ) 2 ( 1 2 1 0 0 1 0 0 4 2 1 2 3 0 0 1 2 2 0 4 2 2 1 } ]{ [ X X X X X X X I A 0 ) 2 ( 3 = X We assume 1 ) 2 ( 1 = X 0 2 ) 2 ( 3 ) 2 ( 2 ) 2 ( 1 = + X X X 0 2 1 ) 2 ( 2 = + X 5 . 0 ) 2 ( 2 = X = = ) 3 ( 3 ) 3 ( 2 ) 3 ( 1 ) 3 ( 3 ) 3 ( 2 ) 3 ( 1 3 0 0 0 1 1 0 4 2 2 3 3 0 0 1 3 2 0 4 2 3 1 } ]{ [ X X X X X X X I A We assume 1 ) 3 ( 3 = X 0 ) 3 ( 3 ) 3 ( 2 = + X X 0 1 ) 3 ( 2 = + X 1 ) 3 ( 2 = X 0 4 2 2 ) 3 ( 3 ) 3 ( 2 ) 3 ( 1 = + X X X 0 4 2 2 ) 3 ( 1 = + X 1 ) 3 ( 1 = X
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2 The eigenvector matrix is: = 1 0 0 1 5 . 0 0 1 1 1 ] [ X We normalize the columns of [ X ] to be unit vectors: = 0.57735 0 0 0.57735 0.447213 0 0.57735 0.894427 1 ] [ X b. 3 4 1 1 1 ] det[ = × × = B 0 2 3 4 2 1 4 ) 1 )( 1 ( 1 1 4 1 ] det[ 2 2 = + = + = = = λ I A 0 3 2 2 = 0 ) 1 )( 3 ( = + Eigenvalues of ] [ A are: 3 and -1 Eigenvectors: = = ) 1 ( 2 ) 1 ( 1 ) 1 ( 2 ) 1 ( 1 1 1 1 2 1 4 2 1 1 4 1 } ]{ [ X X X X X I A We assume 1 ) 1 ( 1 = X 0 2 ) 1 ( 2 ) 1 ( 1 = X X 0 2 1 ) 1 ( 2 = X 5 . 0 ) 1 ( 2 = X = = ) 2 ( 2 ) 2 ( 1 ) 2 ( 2 ) 2 ( 1 2 2 2 2 1 4 2 1 1 4 1 } ]{ [ X X X X X I A We assume 1 ) 2 ( 1 = X 0 2 ) 2 ( 2 ) 2 ( 1 = + X X 0 2 1 ) 2 ( 2 = + X 5 . 0 ) 2 ( 2 = X The eigenvector matrix is: = 5 . 0 5 . 0
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ME581_Assignment04-Sol - ME 581 Assignment #4 Eigenvalue...

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