CHAPTER 04.01 - 04.05: PRIMER ON SIMULTANEOUS LINEAR EQUATIONS: What is a Diagonally Dominant Matrix
Let's go ahead and look at what a diagonally dominant matrix is. In this segment, we will talk about what a diagonally dominant matrix is. And this diagonally dominant matrix plays a very prominent role in using . . . using iterative methods to solve simultaneous linear equations. This also plays a prominent role in knowing whether a particular matrix will have an inverse.
If a matrix is strictly diagonally dominant, then it does have an inverse of the matrix does exist, but let's . . . let's see what a diagonally dominant matrix is. If you have an n by n square matrix is diagonally dominant if the absolute value of the diagonal element, so you're going to look at the diagonal element of each and every row, the absolute value of the diagonal element has to be greater than or equal to the summation of the absolute value of the rest of the elements of that particular row, so you're going to take all the elements of that particular row, except for the diagonal element, so you're going to add them from 1 to n, i not equal to j . . . j not equal to i, because you're not going to include the diagonal element, so the absolute value of the diagonal element has to be greater than the absolute value of each of the elements added together in that particular row, and this has to be true for all . . . all rows, this has to be true for all rows. And then, also, what you need is, you need aii, which is the diagonal element, to be greater than, strictly greater than, the absolute value addition of all the elements in each row . . . in that row, for at least one i. So for one of the rows, you need the strictly greater than equality to be . . . inequality to hold good. You cannot have greater than or equal to in each and every . . . each and every row. So the absolute value of the diagonal element has to be greater than or equal to the absolute value of the individual elements in that row being added together, except for the diagonal element, and that has to be true for each and every row, then you have to have one of the rows where it's strictly greater than.
So let's go ahead and look at an example, and that will make it much more clear to see what we mean by that. Let's suppose somebody gives me a 3 by 3 matrix, and asks me, hey, is this diagonally dominant? So let's suppose somebody gives me 15, 6, 7, 2, -4, -2, 3, 2, and 6, is this diagonally dominant? So what I'll do is I'll take the first row, first column, so I'll take the diagonal element of the first row, take the absolute value of that, which is 15, and see what the . . . what the absolute value of the rest of the elements is, 15, 6, and 7, right? So 6 and 7 are the other elements, I'll take the absolute value of 6, absolute value of 7, because those are the rest of the elements in that particular row, and that gives me 13, and that is greater than or equal to that quantity here. Same here, what is the second row? Second row, the absolute value of the diagonal element is absolute value of -4, which is simply 4, and what are the rest of the elements? The rest of the elements are absolute value of 2 and absolute value of -2, 2 and absolute value of -2, and that gives me 4, and that is greater than or equal to, that . . . that inequality is satisfied. Then if you look at the last . . . last row here, the absolute value of the diagonal element is what, 6, and is that greater than the absolute value of 3 plus the absolute value of 2, which is the rest of the elements, which I get 5, and that is 6, and that is greater than or equal to 5, so I have 15 greater than 13, 4 greater than or equal to 4, 6 greater than or equal to 5, but is . . . but this does not make it diagonally dominant. What I have to also show is that for at least one of the rows, the inequality is strictly greater than, which is true, because here 15 is strictly greater than 13, or here 6 is strictly greater than 5, so A is diagonally dominant. And the reason why that is so is because the absolute value of the diagonal element is greater than the absolute value sum of the rest of the elements, and it is strictly greater than for at least one row, in this case, it is strictly greater than for two rows, so that makes it diagonally dominant.
Let's take another example, let's suppose if we had A equal to, like this, 15, 6, 7, then we had 2, -3, and 2, then 3, 2, and 6. And you can very well see that this one is not diagonally dominant, because, in the first case, absolute value of 15, which is 15, is greater than the absolute value of 6 plus the absolute value of 7, which is 13, that is true for the first row. But, for the second row, the absolute value of -3, which is 3, is it greater than the absolute value of 2 here plus the absolute value of 2 here, which is 4, is it greater than or equal to? No, it is not. So, since it doesn't satisfy the condition for that particular row, then it cannot be diagonally dominant, so A is not diagonally dominant. Let's look at yet another example and see this one. We have A equal to 15, 6, and 9, 2, -4, -2, 3, -2, and 5. Is this diagonally dominant? No, it is not diagonally dominant, why? Because you have 15, absolute value of 15 is equal to 15, is greater than or equal to the absolute value of 6 plus the absolute value of 9, which is 15, that is true. In . . . for the second row here, the absolute value of -4, which is my diagonal element, which is 4, is it greater than the absolute value . . . greater than or equal to the absolute value of the other elements, which is absolute value of 2 and absolute value of -2, which is 4? That is true. Same thing here, what is the absolute value of the diagonal element? It is 5, is it greater than or equal to the absolute value of this diagonal . . . this other element in that particular row, which is 3, plus the other element, which is -2, which is 5? Yes it is, but what you are finding out is that you don't have any row where the strictly greater than inequality is met, because 15 is equal to 15, 4 is equal to 4, 5 is equal to 5, but your strictly greater than inequality is not met anywhere, so A is not diagonally dominant.
Now, we have another part of the diagonally dominant matrix, which is called strictly diagonally dominant, something called strictly diagonally dominant. If you remember the definition of the diagonally dominant matrix, we talked about that you have to have the greater than or equal to inequality met for all the rows, and the greater than inequality met for just at least one row. In the strictly dominant matrix, what you have to have is that the absolute value of the diagonal element has to be strictly greater than the sum of the absolute values of the rest of the elements of that particular row for each and every row, not just for one row, but for each and every row. So you don't have the greater than . . . greater than or equal to inequality anymore, you just have a strictly greater than or equal to . . . greater than inequality, which has to be met for each and every row, only then the particular matrix is strictly diagonally dominant. The reason why we do need to talk about strictly diagonally dominant matrices is of the examples is that, or one of the applications is that if you have a strictly diagonally dominant matrix, the inverse of the matrix exists, you don't have to do any kind of testing for that. And also, so let's go ahead and see . . . see an example of this, so if you have something like this, 15, 3, -9, 6, -9, -2.9, and 8, 3, and 11.1, let's suppose, so let's suppose you have a matrix like this, is this strictly diagonally dominant? Yes it is, because absolute value of 15 is equal to 15 is greater than absolute value of 3 plus absolute value of 9, which is 12, so we're looking at the first row here. Now, when we look at the second row, we'll take the absolute value of the diagonal element, which is -9, which is 9, is it greater than 6 plus the absolute value of -2.9, which is 8.9? Yes it is, that inequality holds good. And the last one is absolute value of the diagonal element of the third row is 11.1, and is it greater than the sum of the absolute value of the rest of the elements? 8 plus 3, which is 11, yes it is, so you have the greater than inequality is met for each and every row, not just one row, but each and every row, so that makes this particular matrix to be . . . this particular matrix, A, here, to be A is strictly diagonally dominant. And that's the end of this segment. |