eig_2009_09_28_02

eig_2009_09_28_02 - 5-1 Eigenvectors S Lall Stanford...

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5 - 1 Eigenvectors S. Lall, Stanford 2009.09.28.02 5. Eigenvectors Eigenvectors and eigenvalues Properties Scaling interpretations Diagonalization
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5 - 2 Eigenvectors S. Lall, Stanford 2009.09.28.02 Eigenvectors and eigenvalues λ C is an eigenvalue of A C n × n if X ( λ ) = det( λI A ) = 0 equivalent to: there exists nonzero v C n s.t. ( λI A ) v = 0 , i.e. , Av = λv any such v is called an eigenvector of A (associated with eigenvalue λ ) there exists nonzero w C n s.t. w T ( λI A ) = 0 , i.e. , w T A = λw T any such w is called a left eigenvector of A
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5 - 3 Eigenvectors S. Lall, Stanford 2009.09.28.02 Properties if v is an eigenvector of A with eigenvalue λ , then so is αv , for any α C , α negationslash = 0 even when A is real, eigenvalue λ and eigenvector v can be complex when A and λ are real, we can always find a real eigenvector v associated with λ : if Av = λv , with A R n × n , λ R , and v C n , then A Re v = λ Re v, A Im v = λ Im v so Re v and Im v
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