CHAPTER 09.03: ADEQUACY OF
SOLUTIONS: Row sum norm of a matrix Example
In
this segment we will take an example of our row-sum norm of a matrix. How do
we calculate it? Let’s suppose somebody says hey, find a row-sum norm of this
matrix. A is given as follows. So you have 10 minus seven, zero. Minus 3,
2.099, 6 and 5 minus 1, 5. So we have already looked at what the definition
of the row-sum norm is, is that the norm of A the infinity norm of the
row-sum norm is given by maximum of 1 less than I less than or equal to M? J
is equal to 1 to N absolute value AIJ for a matrix
which is M row and N columns. So if your M rows and
N columns then the infinity matrix is defined by that. So
if we try to substitute the value of M which is 3rows and 3 columns. So M is
3 and N is 3. ? J has a 1 to 3 absolute value of AIJ. So what basically that
means is that we are going to find the max of all these summations. So we are
going to put I equal to 1, find the summation I equal to two. Find this summation
and so the same for 3. Then we are going to take the maximum of those. So
what that means is that the max of the-taking the I
equal to 1- which is the first row and taking the summations of absolute
value of all elements. So it would be 10 plus absolute value minus seven plus
absolute value of zero. So that is the first row. Then we put I equal to 2
which means that is the 2nd row. So we have to take the absolute value of
each element and add them up together, which would be absolute value minus 3
plus absolute value of 2.099 plus the absolute value of 6. And then you are
going to take the 3rd row, which is I equal to 3, take the absolute value of
each element and add them up together. So that will be absolute value of 5 plus
absolute value of minus one plus absolute value of 5. So
those are the individual summations which we are going to calculating. And
these summations, each of these are max of, the first one would be 17 the next
one will be 11.099 and the last one will be 11. And the max of this is 17.
That is the row-sum norm of that particular matrix. And that’s the end of
this segment. |