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Macroeconomics Exam Review 131

Macroeconomics Exam Review 131 - c 2001 Michael Carter All...

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3.90 By linearity 𝑓 ( x ) = 𝑗 𝑥 𝑗 𝑓 ( x 𝑗 ) 𝑄 defines a quadratic form since 𝑄 ( x ) = x 𝑇 𝑓 ( x ) = ( 𝑖 𝑥 𝑖 x 𝑖 ) 𝑇 𝑗 𝑥 𝑗 𝑓 ( x 𝑗 ) = 𝑖 𝑗 𝑥 𝑖 𝑥 𝑗 x 𝑇 𝑖 𝑓 ( x 𝑗 ) = 𝑖 𝑗 𝑎 𝑖𝑗 𝑥 𝑖 𝑥 𝑗 by Exercise 3.69. 3.91 Let 𝑓 be the symmetric linear operator defining 𝑄 𝑄 ( x ) = x 𝑇 𝑓 ( x ) By the spectral theorem (Proposition 3.6), there exists an orthonormal basis x 1 , x 2 , . . . , x 𝑛 comprising the eigenvectors of 𝑓 . Let 𝜆 1 , 𝜆 2 , . . . , 𝜆 𝑛 be the corresponding eigenvalues, that is 𝑓 ( x 𝑖 ) = 𝜆 𝑖 x 𝑖 𝑖 = 1 , 2 . . ., 𝑛 Then for x = 𝑥 1 x 1 + 𝑥 2 x 2 + ⋅ ⋅ ⋅ + 𝑥 𝑛 x 𝑛 𝑄 ( x ) = x 𝑇 𝑓 ( x ) = ( 𝑥 1 x 1 + 𝑥 2 x 2 + ⋅ ⋅ ⋅ + 𝑥 𝑛 x 𝑛 ) 𝑇 𝑓 ( 𝑥 1 x 1 + 𝑥 2 x 2 + ⋅ ⋅ ⋅ + 𝑥 𝑛 x 𝑛 ) = ( 𝑥 1 x 1 + 𝑥 2 x 2 + ⋅ ⋅ ⋅ + 𝑥 𝑛 x 𝑛 ) 𝑇 ( 𝑥 1 𝑓 ( x 1 ) + 𝑥 2 𝑓 ( x 2 ) + ⋅ ⋅ ⋅ + 𝑥 𝑛 𝑓 ( x 𝑛 )) = ( 𝑥 1 x 1 + 𝑥 2 x 2 + ⋅ ⋅ ⋅ + 𝑥 𝑛 x 𝑛 )
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