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204PS52009

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

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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 = 0 @ 1 2 0 0 2 0 ° 2 ° 2 ° 1 1 A ; B = ° 1 1 0 2 ± ; and C = 0 @ 1 0 0 0 2 0 0 0 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 ² 0 . A is said to be positive de°nite if for all non-zero vectors x 2 R n , x T Ax > 0 . 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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204PS52009 - Problem Set 5 Econ 204 Due Friday August 14...

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