hw1_fa09

hw1_fa09 - UC Berkeley Department of Electrical Engineering...

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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 pre-requisites 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 non-negative. Problem 1.2 Consider a symmetric matrix M in block-partitioned 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 > 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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hw1_fa09 - UC Berkeley Department of Electrical Engineering...

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