Lect25b - Chapter 6 Eigenvalues Eigenvectors Diagonalization Math1111 Eigenvalues Eigenvectors Motivation Recall the problem in last chapter Let A be an

# Lect25b - Chapter 6 Eigenvalues Eigenvectors...

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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:LetAbe ann×nmatrix.Can we find a (nonsingular) matrixSsuch thatS-1AS=λ1λ2...λn?If we could,what are the numbers ofλ1,···,λn?
Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Eigenvalues & Eigenvectors Motivation Recall the problem in last chapter:LetAbe ann×nmatrix.Can we find a (nonsingular) matrixSsuch thatS-1AS=λ1λ2...λn?If we could,what are the numbers ofλ1,···,λn?The numberλimust satisfyASei=λiSei.Note thatSei=0,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