Lect25b - Chapter 6. Eigenvalues & Eigenvectors,...

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Unformatted text preview: Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Motivation Recall the problem in last chapter: Let A be an n × n matrix. Can we find a (nonsingular) matrix S such that S- 1 AS =       λ 1 λ 2 . . . λ n       ? Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Motivation Recall the problem in last chapter: Let A be an n × n matrix. Can we find a (nonsingular) matrix S such that S- 1 AS =       λ 1 λ 2 . . . λ n       ? If we could , what are the numbers of λ 1 , ··· , λ n ? Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Motivation Recall the problem in last chapter: Let A be an n × n matrix. Can we find a (nonsingular) matrix S such that S- 1 AS =       λ 1 λ 2 . . . λ n       ? If we could , what are the numbers of λ 1 , ··· , λ n ? The number λ i must satisfy AS e i = λ i S e i . Note that S e i 6 = , a nonzero vector. Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Definition Definition Let A be a square matrix of order n . If λ is a scalar and x is a nonzero vector such that A x = λ x , then we call λ an eigenvalue of A and x an eigenvector belonging to λ . Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Definition Definition Let A be a square matrix of order n . If λ is a scalar and x is a nonzero vector such that A x = λ x , then we call λ an eigenvalue of A and x an eigenvector belonging to λ ....
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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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Lect25b - Chapter 6. Eigenvalues & Eigenvectors,...

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