ECE 493 HW #2 – Version 1.00 February 24, 2011 Spring 2011 Univ. of Illinois Due Thur, Mar. 10 Prof. Allen Topic of this homework: LinearAlgebra(Cauchy Schwartz inequality for n dimensional space. Eigenvalues and eigenvectors) Deliverables: Show your work. Numerical results are not suFcient expect when speci±cally requested. 1. Prove the Cauchy Schwartz inequality by following the argument below. (a) Use the argument based on the proof on pages 424-426 in the text book. However use the following. De±ne E ( a ) = || V-aU || and then ±nd the minimum of E with respect to a . As shown in the book, use this to prove the CS inequality (13) page 424. (b) Give a diagram showing what is happening when you minimize E ( a ) with respect to a . Provide a diagram showing the minimum value of E ( a * ) where a * is the value such that the gradient of E with respect to a is zero. 2. (a) Prove the triangular inequality || vu + v± || ≤ || vu || + || v± || . All of these are to be done for
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