eig_2009_09_28_02

eig_2009_09_28_02 - 5 1 Eigenvectors S Lall Stanford...

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Unformatted text preview: 5 - 1 Eigenvectors S. Lall, Stanford 2009.09.28.02 5. Eigenvectors • Eigenvectors and eigenvalues • Properties • Scaling interpretations • Diagonalization 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 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...
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

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eig_2009_09_28_02 - 5 1 Eigenvectors S Lall Stanford...

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