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Chapter6 - Chapter 6 Eigensystems Keith E Emmert...

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Unformatted text preview: Chapter 6: Eigensystems Keith E. Emmert Eigenvalues and Eigenvectors Chapter 6: Eigensystems Keith E. Emmert Tarleton State University January 15, 2010 Chapter 6: Eigensystems Keith E. Emmert Eigenvalues and Eigenvectors Overview Eigenvalues and Eigenvectors Chapter 6: Eigensystems Keith E. Emmert Eigenvalues and Eigenvectors Definition Definition 1 Let A be any matrix, real or complex. A number λ is an eigenvalue of A if the equation Ax = λ x is true for some a58a58a58a58a58a58a58a58 nonzero a58a58a58a58a58a58a58 vector x . The vector x is an eigenvector associated with the eigenvalue λ . Remark 2 Eigenvalues and eigenvectors may be complex. Definition 3 Let L be a linear operator mapping a vector space V into itself. If L v = λ v and v negationslash = , then we call λ an eigenvalue and v an eigenvector of L . Chapter 6: Eigensystems Keith E. Emmert Eigenvalues and Eigenvectors A Word about Complex Eigenvalues Theorem 4 If A is a real matrix and has a complex eigenvalue λ , then the conjugate, λ , is also an eigenvalue and Ax = λ x and A x = λ x . Chapter 6: Eigensystems Keith E. Emmert Eigenvalues and Eigenvectors Theory and the Characteristic Equation Theorem 5 A scalar λ is an eigenvalue of a matrix A if and only if Det (...
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Chapter6 - Chapter 6 Eigensystems Keith E Emmert...

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