Week9 - ECON 117 Mathematical Economics Spring 2008 Week 9...

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ECON 117 Spring 2008 Mathematical Economics Week 9 4.2 Properties of eigenvalues and eigenvectors Let be symmetric () nn × A (1) Eigenvalues are “real” Page 1 of 8
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ECON 117 Spring 2008 Mathematical Economics (2) The two eigenvectors obtained from two different eigenvalues , n ij xx \ , λλ are orthogonal, i.e., =0 i j (3) If an eigenvalue λ has multiplicity , then there can be k orthogonal eigenvectors from k . Page 2 of 8
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ECON 117 Spring 2008 Mathematical Economics (4) If has an order , then the number of different eigenvalues may be less than but the number of distinct eigenvectors that are mutually orthogonal is n . A ( nn × ) n (5) Diagonalization of : A ′⋅ ⋅= XAX Λ Page 3 of 8
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ECON 117 Spring 2008 Mathematical Economics (6) () 1 n i i tr λ = = A (7) 1 n i i = = A (8) ρ = A the number of non-zero eigenvalues (9) eigenvalues { A } 12 ,,, n λλ , eigenvectors { } n xx x Then, 2 A eigenvalues { } 2 n , eigenvectors { } n x Page 4 of 8
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ECON 117 Spring 2008 Mathematical Economics (10) eigenvalues { A } 12 ,,, n λ λλ eigenvectors { } n xx x
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This note was uploaded on 09/15/2008 for the course ECON 324 taught by Professor Kim during the Spring '08 term at HKUST.

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Week9 - ECON 117 Mathematical Economics Spring 2008 Week 9...

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