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M235A3 - MATH 235 Eigenvalues Eigenvectors Assignment 3...

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MATH 235 Eigenvalues & Eigenvectors Assignment 3 Hand in questions 3,5,8,9(a,b),10,12 by 9:30 am on Wednesday October 11, 2006. 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 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 2 × 2 matrix of the form A = a c b d where a, b, c, d are positive numbers such that a + b = c + d = 1. Verify that c b and 1 - 1 are eigenvectors of A . What are the corresponding eigenvalues? 2. Let a, b R , b = 0, and let A = a - b b a . Find the eigenvalues and eigenvectors of A . 3. Suppose a real 3 × 3 matrix A has the real eigenvalue 2 and two other eigenvalues. Also suppose that tr( A )=8 and det( A ) = 50. Find the other two eigenvalues of A . 4. A 3 × 3 matrix B is known to satisfy tr( B ) = - 1, det( B ) = 12 and one eigenvalue is 3. (a) Find all the eigenvalues of B and their algebraic multiplicities. (b) Express B 3 as a linear combination of I 3 , B and B 2 . (c) Find the eigenvalues of B - 1 . (d) Find tr( B - 1 ). 5. Using induction or otherwise, prove that the characteristic equation of the n × n matrix C = 0 0 · · · 0 - a 0 1 0 · · · 0 - a 1 0 1 .
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