Fall2011-HW6-soln - Homework 6 Solution Sketches Math 371...

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Homework 6 Solution Sketches Math 371, Fall 2011 Assigned: Thursday, October 20, 2011 Due: Thursday, October 26, 2011 Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Norms) (a) Do problems #3(a), (b) from page 180 (b) Consider the matrix A = 2 0 0 0 3 4 0 2 - 1 . Compute the spectrum of A T A . What is the 2 matrix norm of A ? (c) Compute the matrix norm of A . Find a vector x such that A = Ax / x . (d) Verify your calculations in (b) and (c) using Matlab. We prove that || · || 1 is a vector norm as follows. It is clear that x 1 0 . If x 1 = 0 , then | x i | = 0 for each i . So x = 0 . λx = | λ | | x i | = | λ | | x i | . x + y 1 | x i | + | y i | = x 1 + y 1 . #3(b). The following 1 norms are easily determined. (3 , - 5 , 2) 1 = 9 . 4142 . (2 , 1 , - 3 , 4) 1 = 10 . (4 , - 8 , 1) 1 = 13 . ( e, π, - 1) 1 = 6 . 8599 . The eigenvalues of A T A are 4, 4.8020, and 25.1980. The 2 matrix norm of A is 5 . 0198 . (2) (Condition numbers) Let A be the n × n matrix of the form 2 - 1 - 1 2 - 1 - 1 2 . . . . . . . . . - 1 - 1 2 . Read the help of the Matlab command
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