CS205A hw6_solutions

Scientific Computing

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CS205 Homework #6 Solutions Problem 1 1. Let A be a symmetric and positive definite n × n matrix. If x , y R n prove that the operation x , y A = x T Ay = x · Ay is an inner product on R n . That is, show that the following properties are satisfied (a) u + v , z A = u , z A + v , z A (b) α u , v A = α u , v A (c) u , v A = v , u A (d) u , u A 0 and equality holds if and only if u = 0 2. Which of those properties, if any, fail to hold when A is not positive definite? Which fail to hold if it is not symmetric? Solution 1. (a) u + v , z A = ( u + v ) T Az = u T Az + v T Az = u , z A + v , z A (b) α u , v A = ( α u ) T Av = α ( u T Av ) = α u , v A (c) u , v A = u T Av = u T A T v = Au · v = v · Au = v T Au = v , u A by symmetry (d) u , u A = u T Au 0 if u = 0 by positive definiteness and equality holds trivially when u = 0. 2. Property (3) holds if and only if A is symmetric. Property (4) holds if and only if A is positive definite (by definition) Problem 2 1. Let x 1 , x 2 , . . . , x k be an A -orthogonal set of vectors, that is x T i Ax j = 0 for i = j . Show that if A is symmetric and positive definite, then x 1 , x 2 , . . . , x k are linearly independent. Does this hold when A is symmetric but not positive definite? 2. Let x 1 , x 2 , . . . , x n be n linearly independent vectors of R n and A a n × n symmetric positive definite matrix. Show that we can use the Gram-Schmidt algorithm to create a full A -orthogonal set of n vectors. That is, subtracting from
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