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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 textbooks definition of the characteristic polynomial of an n n matrix A , p A ( ), is p A ( ) := det( A I n ). Another commonly used definition is c A ( ) := det( I n A ). These two polynomials are closely related through p A ( ) = ( 1) n c A ( ). 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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