The following question explores matrix norms and condition numbers :

{a} For the following matrix given by A=E :1]. calculate ||x1||1+ \"Aug. ||.-'1||.3.u as well as the condition numbers for each norm by hand. Is A well or illconditioned? {b} Recall the formulas from Theorems 2.? and 2.3 in the text book. If you assume that taking the absolute value and determining the maximum does not contribute to the overall computational cost. how many flops {floating point operations} are needed to calculate ||A||1 and ||A||gD for A E 11\"\"? By what factor \"will the calculation time increase when you double the size of matrix size? {c} How implement a simple code that calculates \"AMI and ||.r-].||m for a matrix of any size n 3 1. Try to do this without using loopsll Using system sizes of m = l. \"rm 2 Enhk 2 1I . . .,'?. determine how long your code talces2 to calculate ||A||1 and ||A||m for a matrix A E Rmx'\" with random entries and report the results. Can you conrm the estimate from {b}? {d} {extra credit} MATLAB has the buildin function norm to calculate matrix norms.3 Calculate for the system sizes in {c} \"AHI and ||.r-].||m using both your implementation and MATLABis norm function+ determine for each by how long each code takes and plot the results in one graph. Cm average. by what factor is MATLAE-is implementation faster than yours? Please also hand in your code. Theorem ?. ? The matrix norm subordinate to the vector norm .|20 can be expressed, for an "* " matrix A _ 1ajj hajj_ ER^, 215 11 A|60 = May 12.39 ) 1 = 1 Ilanil . 1 = 1 This result is often loosely expressed by saying that the Do- nom of a matrix is its largest Foul - build.Theorem 2.8 The matrix norm subordinate to the vector norm | . | 1 can be expressed, for an n x n matrix A = (@ij)i