Macroeconomics Exam Review 131

Macroeconomics Exam - 3.90 By linearity x = ∑ x defines a quadratic form since x = x x = ∑ x ⎛ ⎝ ∑ x ⎞ ⎠ = ∑ ∑ x x = ∑ ∑ by

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Unformatted text preview: 3.90 By linearity ( x ) = ∑ ( x ) defines a quadratic form since ( x ) = x ( x ) = ( ∑ x ) ⎛ ⎝ ∑ ( x ) ⎞ ⎠ = ∑ ∑ x ( x ) = ∑ ∑ by Exercise 3.69. 3.91 Let be the symmetric linear operator defining ( x ) = x ( x ) By the spectral theorem (Proposition 3.6), there exists an orthonormal basis x 1 , x 2 ,..., x comprising the eigenvectors of . Let 1 , 2 ,..., be the corresponding eigenvalues, that is ( x ) = x = 1 , 2 ..., Then for x = 1 x 1 + 2 x 2 + ⋅⋅⋅ + x ( x ) = x ( x ) = ( 1 x 1 + 2 x 2 + ⋅⋅⋅ + x ) ( 1 x 1 + 2 x 2 + ⋅⋅⋅ + x ) = ( 1 x 1 + 2 x 2 + ⋅⋅⋅ +...
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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