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 is 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 for at least one I. So
for at least one row this particular inequality is strictly
greater than. That's when we will consider A matrix to be diagonally dominate. So let's
look at an example and see whether a particular matrix is
diagonally dominated or not. So what we are going to do is,
we are going to take several examples and to 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 an 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 1 2 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 will be 6.
Let's see whether 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 to be greater than
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 just for one row. So this
particular matrix is diagonally dominant. 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,
plus absolute value 1 which is 6. And 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
the 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.