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Unformatted text preview: UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Problem Set 1 Fall 2009 Issued: Thursday, August 27 Due: Tuesday, September 8, 2009 The first part of this problem set provides some practice on the mathematical prerequisites for this course (vector calculus, elementary analysis, and linear algebra) and will calibrate your degree of preparation for the course. Reading: Boyd and Vandenberghe, 3.1, 3.2 and Appendix A for background material. Problem 1.1 The symmetric matrix Q R n n is positive semidefinite means that x T Qx 0 for all x R n . Prove that Q is positive semidefinite if and only all of its eigenvalues are nonnegative. Problem 1.2 Consider a symmetric matrix M in blockpartitioned form M = bracketleftbigg A B B T C bracketrightbigg . Prove that the following two statements are equivalent: (a) M is strictly positive definite (i.e., x T Mx &gt; 0 for all x negationslash = 0) (b) A is strictly positive definite, and C B T A 1 B is strictly positive definite....
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 Spring '11
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 Electrical Engineering, Derivative, Convex function, Symmetric matrix, x∗ ∈ Rn

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