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 that’s 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. |