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# assign3 - MATH 235/W08 Eigenvalues& Eigenvectors...

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Unformatted text preview: MATH 235/W08 Eigenvalues & Eigenvectors Assignment 3 due by 9:30 am on Wed. Feb. 6/08. Notes: 1) The trace of an n × n matrix A = [ a ij ], denoted tr( A ), is defined to be the sum of the diagonal elements, that is, tr( A ) = ∑ n i =1 a ii . 2) Our textbook’s definition of the characteristic polynomial of an n × n matrix A , p A ( λ ), is p A ( λ ) := det( A − λI n ). Another commonly used definition is c A ( λ ) := det( λI n − A ). These two polynomials are closely related through p A ( λ ) = ( − 1) n c A ( λ ). 1. Consider a matrix of the form 2 a − c a − c − 2 a + 2 c 2 a − 2 b a − 2 a + 2 b 2 a − b − c a − c − 2 a + b + 2 c (a) Verify that the two vectors ( 1 1 1 ) T and ( 1 1 ) T are eigen- vectors and find the corresponding eigenvalues. (b) Find a third eigenvalue and a corresponding eigenvector. 2. Let a, b ∈ R , b negationslash = 0, and let A = bracketleftbigg a − b b a bracketrightbigg . Find the eigenvalues and eigenvectors of A . 3. For the matrices A = parenleftbigg 1 + 3 i − 4 − 2 1 − 3 i parenrightbigg , B = − 2 − 3 − 1 1 − 1 2 2 5 : (i) Find the characteristic polynomials of A and B . (ii) Find the eigenvalues of A and B ....
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assign3 - MATH 235/W08 Eigenvalues& Eigenvectors...

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