Problem 2 (30 pts). In the following, let A E Rmxm be a diagonal matrix with nonzero diagonal entries (1,3, and let ||H represent an arbitrary pnorm. a) What is the matrix condition number MA) relative to the matrix norm induced by \"H? Give your answer in terms of entries ai. b) We showed in class that the matrix multiplication problem b : Ax has a relative error bound: ll5bl| lI5Xl| Mb\" 3 \"(A) IIXII V" i 0' Determine an input x and perturbation 6x for which this bound is tight (||6b|| = rc(A) ||bi| ||6x|| / HxH). Hint: how can you pick unit x and 6x to make b and 6b as small and large as possible, respectively? You'll probably nd imm := arg'mini |a,-|, imax := argmaxi Iail helpful. c) Similarly, we saw that, when b is represented exactly but A is nonexaet, the solution to Ax = b has a relative error bound: ||6XI| NM\" 2 g H, A + 0 6A , \"x\" ( ) \"A\" (l l ) Stated more precisely, applying the change of variables 6A : th for a unit perturbation HMH = Determine a right-hand side b and a perturbation direction 671 for which this bound is tight. Hint: pick b and 6A such that x and 6x have a single nonzero entry at index i. You should then be able to obtain a simple expression for the limit that depends on i. Finally, pick thei that maximizes this expression. Below is reference showed in class! Condition Numbers: Matrix Multiplication 0 Important example: consider matrix multiplication f(x) = Ax for A 6 CW" 0 In this case Vf = A 0 Let H x II, lle|| denote arbitrary norms for C" and C\