CHAPTER 09.02: ADEQUACY OF SOLUTIONS: Ill-conditioned and well-conditioned system of equations



In this segment we will talk about how to find the norm of a matrix. So for a M by N matrix, A. We are going to define the infinity norm or what is called the row-sum norm. So this particular norm is called the infinity norm. It is also called the row-sum norm. These are some of the names for the same norm. But again what you have to understand is that there are different types of norms. You might have a column-sum norm, row-sum norm and so on and so forth. But in this particular course we are limiting ourselves to the infinity norm. Only because of that fact that we need to use the norms to do other things.


But if you are interested in finding out what other norms are, you can always do that. So what is the definition of the infinity norm? It is as follows. It is a max. 1 less than or equal to I less than or equal to M? J has a 1 to N, absolute value AIJ. So what basically that means is that you are going to find this sum right here. Going from the first column to the Nth column, for each of the rows. So when we say 1 less than or equal to I less than or equal to M means that I will take the value of 1,2,3,4,5,6..all the way up to M and you are going to find this summation for each value of I. Then once you have done that you are going to find what is the maximum for all those summations and the way the summations have to be taken is that you take the absolute value of each element in that particular row and add them all up.


This would be clear from example but thatís what this whole row-sum norms mean. Again mind that the norms of the matrixes are defined from a rectangular matrix not only for a square matrixes but rectangular matrixes the norms defined. Although in this course we are only going to calculate norms of square matrixes since were dealing with systems of simultaneous linear equations. So we do need to keep in mind the norm can also be defined for rectangular matrixes. And this is the end of this segment.