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22-eigenvectors - CALCULATING EIGENVECTORS HOMEWORK Section...

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CALCULATING EIGENVECTORS Math 21b, O.Knill HOMEWORK: Section 7.3: 8,16,20,28,42,38*,46* NOTATION. We often just write 1 instead of the identity matrix 1 n . COMPUTING EIGENVALUES. Recall: because λ - A has ~v in the kernel if λ is an eigenvalue the characteristic polynomial f A ( λ ) = det( λ - A ) = 0 has eigenvalues as roots. 2 × 2 CASE. Recall: The characteristic polynomial of A = a b c d is f A ( λ ) = λ 2 - ( a + d ) / 2 λ +( ad - bc ). The eigenvalues are λ ± = T/ 2 ± p ( T/ 2) 2 - D , where T = a + d is the trace and D = ad - bc is the determinant of A . If ( T/ 2) 2 D , then the eigenvalues are real. Away from that parabola in the ( T, D ) space, there are two different eigenvalues. The map A contracts volume for | D | < 1. NUMBER OF ROOTS. Recall: There are examples with no real eigenvalue (i.e. rotations). By inspecting the graphs of the polynomials, one can deduce that n × n matrices with odd n always have a real eigenvalue. Also n × n matrixes with even n and a negative determinant always have a real eigenvalue. IF ALL ROOTS ARE REAL. f A ( λ ) = λ n - tr( A ) λ n - 1 + ... + ( - 1) n det( A ) = ( λ - λ 1 ) ... ( λ - λ n ), we see that i λ i = trace( A ) and Q i λ i = det( A ). HOW TO COMPUTE EIGENVECTORS? Because ( λ - A ) ~v = 0, the vector ~v is in the kernel of λ - A . EIGENVECTORS of a b c d are ~v ± with eigen- value λ ± .
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