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

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Unformatted text preview: 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 = for some x 6 = . i.e. the equation has non-trivial solution ⇔ det ( A- λ I ) = . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A- λ I ) = . (iii) A- λ I is singular. (iv) N ( A- λ I ) 6 = { } . (v) rank ( A- λ I ) < n where A is of order n . (i) ⇔ (ii) ⇔ (iii) ⇔ (iv): Well-known. 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 = for some x 6 = . i.e. the equation has non-trivial solution ⇔ det ( A- λ I ) = . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A- λ I ) = . (iii) A- λ I is singular. (iv) N ( A- λ I ) 6 = { } . (v) rank ( A- λ I ) < n where A is of order n . (iv) ⇒ (v): Rank-Nullity Theorem 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 = for some x 6 = . i.e. the equation has non-trivial solution ⇔ det ( A- λ I ) = . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A- λ I ) = . (iii) A- λ I is singular. (iv) N ( A- λ I ) 6 = { } . (v) rank ( A- λ I ) < n where A is of order n . (v) ⇒ (i): Exercise 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 = for some x 6 = . i.e. the equation has non-trivial solution ⇔ det ( A- λ I ) = . Furthermore, the following are equivalent: (i) λ is an eigenvalue of A . (ii) det ( A- λ I ) = . (iii) A- λ I is singular. (iv) N ( A- λ I ) 6 = { } . (v) rank ( A- λ I ) < n where A is of order n . Remark is always an eigenvalue of a singular matrix. 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 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 ....
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Lect26 - Chapter 6. Eigenvalues & Eigenvectors,...

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