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Macroeconomics Exam Review 129

Macroeconomics Exam Review 129 - c 2001 Michael Carter All...

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3.83 Suppose 𝑓 is singular. Then there exists x = 0 such that 𝑓 ( x ) = 0. Therefore x is an eigenvector with eigenvalue 0. Conversely, if 0 is an eigenvalue 𝑓 ( x ) = 0 x = 0 for any x = 0 . Therefore 𝑓 is singular. 3.84 Since 𝑓 ( x ) = 𝜆 x 𝑓 ( x ) 𝑇 x = 𝜆 x 𝑇 x = 𝜆 x 𝑇 x 3.85 By Exercise 3.69 𝑎 𝑖𝑗 = x 𝑇 𝑖 𝑓 ( x 𝑗 ) 𝑎 𝑗𝑖 = x 𝑇 𝑗 𝑓 ( x 𝑖 ) = 𝑓 ( x 𝑖 ) 𝑇 x 𝑗 and therefore 𝑎 𝑖𝑗 = 𝑎 𝑗𝑖 ⇐⇒ x 𝑇 𝑖 𝑓 ( x 𝑗 ) = 𝑓 ( x 𝑖 ) 𝑇 x 𝑗 3.86 By bilinearity x 𝑇 1 𝑓 ( x 2 ) = x 𝑇 1 𝜆 2 x 2 = 𝜆 2 x 𝑇 1 x 2 𝑓 ( x 1 ) 𝑇 x 2 = 𝜆 1 x 𝑇 1 x 2 = 𝜆 1 x 𝑇 1 x 2 Since 𝑓 is symmetric, this implies ( 𝜆 1 𝜆 2 ) x 𝑇 1 x 2 = 0 and 𝜆 1 = 𝜆 2 implies x 𝑇 1 x 2 = 0. 3.87 1. Since 𝑆 compact and 𝑓 is continuous (Exercises 3.31, 3.62), the maximum is
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