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Lect26 - Chapter 6 Eigenvalues Eigenvectors Diagonalization...

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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Basic Properties I Let A be a square matrix. If λ is an eigenvalue of A , then ( A - λ I ) x = 0 for some x 6 = 0 . i.e. the equation has non-trivial solution det ( A - λ I ) = 0 . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A - λ I ) = 0 . (iii) A - λ I is singular. (iv) N ( A - λ I ) 6 = { 0 } . (v) rank ( A - λ I ) < n where A is of order n . (i) (ii) (iii) (iv): Well-known.
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Basic Properties I Let A be a square matrix. If λ is an eigenvalue of A , then ( A - λ I ) x = 0 for some x 6 = 0 . i.e. the equation has non-trivial solution det ( A - λ I ) = 0 . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A - λ I ) = 0 . (iii) A - λ I is singular. (iv) N ( A - λ I ) 6 = { 0 } . (v) rank ( A - λ I ) < n where A is of order n . (iv) (v): Rank-Nullity Theorem
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Basic Properties I Let A be a square matrix. If λ is an eigenvalue of A , then ( A - λ I ) x = 0 for some x 6 = 0 . i.e. the equation has non-trivial solution det ( A - λ I ) = 0 . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A - λ I ) = 0 . (iii) A - λ I is singular. (iv) N ( A - λ I ) 6 = { 0 } . (v) rank ( A - λ I ) < n where A is of order n . (v) (i): Exercise
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Basic Properties I Let A be a square matrix. If λ is an eigenvalue of A , then ( A - λ I ) x = 0 for some x 6 = 0 . i.e. the equation has non-trivial solution det ( A - λ I ) = 0 . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A - λ I ) = 0 . (iii) A - λ I is singular. (iv) N ( A - λ I ) 6 = { 0 } . (v) rank ( A - λ I ) < n where A is of order n . Remark 0 is always an eigenvalue of a singular matrix.
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Homework 1 Reading Leon (7th edition): p.298-302 Leon (8th edition): p.282-286 Homework 1 Leon (7th edition) Chapter 6 Section 1 Qn. 3-7 Leon (8th edition) Section 6.1 Qn. 3, 4, 6, 8, 9
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Method Method Eigenvalues : Solve det ( A - λ I ) = 0 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 ) = 0 is called a characteristic equation .
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . Find the eigenvalues & the corresponding eigenvectors of A = 3 2 3 - 2 .
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Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Examples Example . Find the eigenvalues & the corresponding eigenvectors of A = 3 2 3 - 2 .
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