204PS52009 - Problem Set 5 Econ 204 Due Friday, August 14...

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Unformatted text preview: Problem Set 5 Econ 204 Due Friday, August 14 Exercise 1 Identify which of the following matrices are diagonalizable and provide the diagonalization. If the diagonalization does not exist, prove it. A = @ 1 2 2 & 2 & 2 & 1 1 A ; B = & 1 1 2 ; and C = @ 1 2 3 1 A Exercise 2 An n n matrix A is called positive semide&nite if for all vectors x 2 R n , x T Ax . A is said to be positive de&nite if for all non-zero vectors x 2 R n , x T Ax > . Suppose A is an n n positive semide&nite matrix. Is B T AB positive semide&nite for an arbitrary n m matrix B ? Suppose A is positive de&nite; is B T AB in this case positive de&nite? What assumptions (if any) can you make to get these conclusions? Exercise 3 Let u;v be vectors in R 2 . a ) Let & & v; & & 2 R be the vector in direction of vector v that minimizes k u & &v k , where k w k denotes the Euclidean norm of w . What can you conclude about the vectors z := u & & & v and v . What is the expression for & & ?...
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at University of California, Berkeley.

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204PS52009 - Problem Set 5 Econ 204 Due Friday, August 14...

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