CHAPTER 01.09: INTRODUCTION: Diagonally dominant matrix

So in this case we will look at what is a diagonally dominant matrix. So N by N matrix A - so it is a square matrix - is diagonally dominant. If what happens is that each of the elements which are on the diagonal - the absolute value of that - will be greater than the sum of the absolute elements of the rest of the elements which are in that row. So we will explain it through example and also we will go through this formula again.

So we have summation here J is equal to 1 to N. I is not equal to J. Absolute value of AIJ, I is equal to 1 through N. And this is greater than or equal to. And then we say AII has to be strictly greater than the summation which we just wrote. J is equal to 1 through N. I not equal to J. AIJ for at least 1 I. So let’s concentrate on this part here. So what this means is that you have a diagonal element. You take the absolute value of that. Then you look at the row I and you take the sum of the absolute elements, the absolute value of all the elements in that row except for AII. That is why I is not equal to J. And then you add all of them up and then you find of out if the diagonal element is greater than or equal to that sum. Now here we see greater than or equal to and here we see greater than, so at least for one of the values of I, it can be 1, 2, 3, N, any of the values of I, for at least one row, that’s what we mean by at least one I. So for at least one row this particular inequality is strictly greater than. That’s when we will consider the A matrix to be diagonally dominate.

So let’s look at an example and see with a particular matrix if it is diagonally dominated or not. So what we are going to do is we are going to take several examples and make sure that is the case. So let’s look at this particular matrix here and see if we can classify it as a diagonally dominated matrix. 15, 6, 7, 2, -4, -2, 3, 2, 6. So here what you are finding out is that, let’s look at row 1. So when I has a 1. Row number 1. The absolute value of the diagonal element is 15. You are finding out right here. So absolute value of A 1 1 is equal to 15. Let’s see what the sum of the rest of the elements are, which is A 1 2 plus A 1 3. The absolute value of the rest of the elements in that row will be the absolute value of A 1 2 and absolute of A 1 3. You are going to add those up - and what do we get? We get the absolute value of 6 plus the absolute value of 7 - that is 13. So it is greater than or equal to.

So let’s go for I is 2. I is equal to 2. The second row, second column absolute value is absolute value -4. Which is 4. Let’s see if it’s greater than or equal to the rest of the elements being added together with their absolute values. So we have A 2 1, which is right here. Absolute value of that plus the absolute value of A 2 3, which is right there. A 2 3. Those are the rest of the elements in the A matrix right there. So this one will be A 21 is 2. Absolute value of 2. A 2 3 is -2. So that gives me 4. And 4 is greater than or equal to 4. So that inequality is satisfied.

Let’s go for I is equal to 3. Let’s go to the 3rd row, 3rd column. That number is 6. Absolute value of 6 is 6. Let’s see if it is greater than or equal to the sum of the absolute value of the elements which are left over. So is it greater than or equal to A 3rd row, 1st column -absolute value- which is this one. Plus A 3rd row 2nd column, which is this one here. Absolute value of 3 plus absolute value of 2 which gives 5. And 6 is greater than or equal to 5.

So we have satisfied the inequality, which we had for the 1st condition and for each row the diagonal element absolute value has been greater than or equal to the sum of the rest of the elements in that particular row. But are we meeting the strictly greater than condition? Yes we are. We are meeting the condition for the 1st row. Not for the 2nd row but for the 3rd row. But we need this strictly greater than inequality to be satisfied for one row. So this particular matrix is diagonally dominant.

So let’s look at this matrix. Somebody gives us a matrix like this. 25, 5, 1, 64, 8, 1, 144, 12, 1. Somebody is saying: hey can you describe if this matrix is diagonally dominant or not? So again what I will do is I will take the absolute value of the first row and first column, which is C 1 1, which is 25 here. And now let me look at the rest of the elements of that particular row, which is this element and this element. So that is the absolute value of the first row, 2nd column plus the absolute value of the 1st row 3rd column, which is 5, absolute value, absolute value 1 which is 6. Surely is greater than or equal to. Let’s look at the 2nd row. This is I equal to 1. That’s what we talked about, I equal to 1. Let’s look at I equal to 2. I equal to 2 will have 2nd row and 2nd column, which is this one right here. The absolute value of 8 is 8. Let’s see if this 8 is greater than or equal to sum of the absolute elements in the rest of the row here. So that will be C 2 1. 2nd row 1st column. And then left is 2nd row 3rd column. Which is absolute value of 64 plus the absolute value of 1, which is 65. IS that greater than or equal to 65? Is 8 greater than or equal to 65? No. So C is automatically not diagonally dominant. I don’t need to look at the 3rd row because it has already violated the inequality for the 2nd row. So it is not diagonally dominant. So that is the end of this segment.