Lect27 - Chapter 6. Eigenvalues & Eigenvectors,...

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Unformatted text preview: Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . The eigenvalues of B = 2- 3 1 1- 2 1 1- 3 2 is 0,1,1 . Check the formula in (iii) & (iv). Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . The eigenvalues of B = 2- 3 1 1- 2 1 1- 3 2 is 0,1,1 . Check the formula in (iii) & (iv). Example . If the characteristic polynomial of A is p ( x ) = (- 1 ) n ( x- λ 1 ) d 1 ( x- λ 2 ) d 2 ··· ( x- λ k ) d k , show that (i) d 1 + d 2 + ··· + d k = n , (ii) trace of A = d 1 λ 1 + d 2 λ 2 + ··· + d k λ k , where n is the order of A . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Theorem 6.1.1 Theorem 6.1.1 Let A and B be n × n matrices. If B is similar to A , then A and B have the same characteristic polynomial. Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Theorem 6.1.1 Theorem 6.1.1 Let A and B be n × n matrices. If B is similar to A , then A and B have the same characteristic polynomial. Proof . If B is similar to A , there is a nonsingular S such that S- 1 BS = A . The characteristic polynomial of A is = det ( A- xI ) Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Theorem 6.1.1 Theorem 6.1.1 Let A and B be n × n matrices. If B is similar to A , then A and B have the same characteristic polynomial. Proof . If B is similar to A , there is a nonsingular S such that S- 1 BS = A . The characteristic polynomial of A is = det ( A- xI ) Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Theorem 6.1.1 Theorem 6.1.1 Let A and B be n × n matrices. If B is similar to A , then A and B have the same characteristic polynomial. Proof . If B is similar to A , there is a nonsingular S such that S- 1 BS = A . The characteristic polynomial of A is = det ( A- xI ) = det ( B- xI ) = characteristic polynomial of B . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Theorem 6.1.1 Theorem 6.1.1 Let A and B be n × n matrices. If B is similar to A , then A and B have the same characteristic polynomial....
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Lect27 - Chapter 6. Eigenvalues & Eigenvectors,...

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