22-eigenvectors

22-eigenvectors - CALCULATING EIGENVECTORS Math 21b,...

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Unformatted text preview: 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 )....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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