Lect26 - Chapter 6. Eigenvalues & Eigenvectors,...

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Unformatted text preview: Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Method Method Eigenvalues : Solve det ( A- λ I ) = for λ Eigenvector : Find nonzero vectors in the nullspace N ( A- λ I ) Note that det ( A- λ I ) is a polynomial in λ and we call p ( λ ) = det ( A- λ I ) the characteristic polynomial of A . det ( A- λ I ) = is called a characteristic equation . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . Find the eigenvalues & the corresponding eigenvectors of A = 3 2 3- 2 ! . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . Find the eigenvalues & the corresponding eigenvectors of A = 3 2 3- 2 ! . Ans . The characteristic equation is 3- λ 2 3- 2- λ = . i.e. λ 2- λ- 12 = . So λ = 4 or- 3 . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . Find the eigenvalues & the corresponding eigenvectors of A = 3 2 3- 2 ! . Ans . The characteristic equation is 3- λ 2 3- 2- λ = . i.e. λ 2- λ- 12 = . So λ = 4 or- 3 . Solving ( A- 4 I ) x = , we get x = α ( 2 1 ) T . Any nonzero multiple of 2 1 ! is an eigenvector belonging to the eigenvalue 4. Eigenvector belonging to the eigenvalue- 3 : β- 1 3 ! , β 6 = Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors (Cont’d) Examples Example . Find the eigenvalues & the corresponding eigenvectors of B =     2- 3 1 1- 2 1 1- 3 2     . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors (Cont’d) Examples Example . Find the eigenvalues & the corresponding eigenvectors of B =     2- 3 1 1- 2 1 1- 3 2     . Ans . The characteristic equation is 2- λ- 3 1 1- 2- λ 1 1- 3 2- λ = . i.e.- λ ( λ- 1 ) 2 = . So λ = or 1 . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors (Cont’d) Examples Example . Find the eigenvalues & the corresponding eigenvectors of B =     2- 3 1 1- 2 1 1- 3 2     . Ans . The characteristic equation is 2- λ- 3 1 1- 2- λ 1 1- 3 2- λ = . i.e.- λ ( λ- 1 ) 2 = . So λ = or 1 . Solving ( A- λ I ) x = , we get Eigenvector corresponding to λ = : α ( 1 1 1 ) T , α 6 = . Eigenvector corresponding to λ = 1 : x = α     3 1     + β    - 1 1     and x 6 = . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors (Cont’d) Examples Example . Find the eigenvalues & the corresponding eigenvectors of C = 1 2- 2 1 !...
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This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect26 - Chapter 6. Eigenvalues & Eigenvectors,...

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