Lect27 - Chapter 6. Eigenvalues & Eigenvectors,...

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Unformatted text preview: Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Diagonalization Motivation Recall our problem again: Find an invertible S such that S- 1 AS = D is a diagonal matrix. Alternative formulation: Let L : R n → R n be defined by L ( x ) = A x . The question is: Can we find a basis F for R n such that the matrix representation of L w.r.t. F is D ? Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Diagonalization Motivation Recall our problem again: Find an invertible S such that S- 1 AS = D is a diagonal matrix. Alternative formulation: Let L : R n → R n be defined by L ( x ) = A x . The question is: Can we find a basis F for R n such that the matrix representation of L w.r.t. F is D ? If we could , what were the vectors in the basis F ? Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Diagonalization Motivation Recall our problem again: Find an invertible S such that S- 1 AS = D is a diagonal matrix. Alternative formulation: Let L : R n → R n be defined by L ( x ) = A x . The question is: Can we find a basis F for R n such that the matrix representation of L w.r.t. F is D ? If we could , what were the vectors in the basis F ? Ans. Eigenvectors. Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Diagonalization Motivation Recall our problem again: Find an invertible S such that S- 1 AS = D is a diagonal matrix. Alternative formulation: Let L : R n → R n be defined by L ( x ) = A x . The question is: Can we find a basis F for R n such that the matrix representation of L w.r.t. F is D ? If we could , what were the vectors in the basis F ? Ans. Eigenvectors. The question becomes: Can we find a basis consisting of eigenvectors? Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Diagonalization Theorem 6.3.1 Theorem 6.3.1 Let A be a square matrix of order n . Suppose λ 1 , λ 2 , ··· , λ k are distinct eigenvalues of A , and x 1 , x 2 , ··· , x k are the corresponding eigenvectors. Then x 1 , x 2 , ··· , x k is linearly independent. Chapter 6. Eigenvalues & Eigenvectors, Diagonalization Math1111 Diagonalization Theorem 6.3.1 Theorem 6.3.1 Let A be a square matrix of order n . Suppose λ 1 , λ 2 , ···...
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Lect27 - Chapter 6. Eigenvalues & Eigenvectors,...

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