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? Letís 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 thatís the end of this segment.