In this segment, we're going to take an example of how you do LU decomposition
of a square matrix.
So this is not an . . . this is not an example of the LU decomposition method
itself, but how do you decompose A is equal to L times
U, because that's what you need in order to be able to do the LU decomposition method,
you first need to decompose your A into L times U.
I mentioned in a separate segment, the way . . . one of the ways to find out A is
equal to L times U is by following the same steps of forward elimination as given
in the Naive Gaussian method.
So if you follow the steps of the Naive Gaussian method, the forward elimination steps,
you'll be able to determine what L and U should be.
So let's go ahead and see that how do we go about doing that through an example.
So let's suppose somebody says that, hey, I'm going to
give you this 3-by-3, three equations, three unknowns, 25, 5, 1, 64,
8, and 1, and 144, 12, and 1.
Now, what I want you to do is, I want you to write it as a lower triangular
matrix multiplied by an upper triangular matrix.
So what we're going to do is, on this we're going to do the forward elimination
steps of Naive Gaussian method, that's what we're
going to do.
So we're going to follow the Naive Gaussian method forward elimination steps
to be able to find out what the L times U of this quantity here is.
So if you recall how forward elimination is done in Naive Gaussian
method is that you want to, in the first step . . . so in the first step, what
you want to do is you want to take this particular row, and make this particular
column here to be 0 below the first row.
So you're taking the first row, and you want to make
this to be 0 and you want to make this to be 0, and in the second step you will try
to make this to be 0, so as to convert this into an upper triangular
matrix, so that's what we're going to do.
So if you look at the first step, the first step . . . so what you have to keep
track of is that what the multipliers are which are going to make this to be 0, this
to be 0, and this to be 0, and that's how we'll be able
to figure out what the L times U is.
So what I'm going to do is, in order to be able to make
this to be 0, I'll have to multiply this row by 64 and divide it by 25.
So first I will divide it by 25, multiply by 64, and subtract it from there.
So if I have the first row, which is 25, 5, and 1, I need to multiply it
by 64 and divided it by 25, which is same as
2.56.
So remember this, that what I'm doing is in order to be able to make
this to be 0, I'll have to take the first row and multiply it by 2.56.
The reason why I'm asking you to remember this is because this is going to show up in
your lower triangular matrix at the place which is trying to make
this to be 0, because what's going to happen is that 64, this is
what we are trying to make 0 by using this multiplier,
so this multiplier is going to show up at second row, first column
in the lower triangular matrix, exactly at the same place where you are trying to
make this to be 0.
So because you are using this first step to make this to be 0,
that's where the multiplier's going to show up in your lower triangular matrix.
So once I multiply this, I get 64, 12.8, and
2.56.
I subtract it from the . . . subtract this from the second row, which is 64, 8,
and 1, and I subtract 64, 12.8, and 2.56.
And if I subtract that, I'll get 0, -4.8, and -1.56, that's what I will get if I subtract
this . . . this row from that row.
So this becomes my second row of the . . . of the coefficient matrix now,
with this being . . . being a 0.
So same thing I'm going to do here, in order to make this to be 0,
I'll have to . . . in order to make this to be 0, I'll have to divide this first row by
25 and multiply by 144, and, again, remember
the multiplier which I'm going to get in order to make this to be 0.
So I have the first row, which is 25, 5, 1, and I'm going to multiply it by 144
divided by 25, which is 5.76.
So when I multiply by 5.76, this number turns out to be 144,
because it should, because I want to make that to be 0, and this is what I
will get for the . . . after I multiply the first row by 5.76.
Now I've got to take my third row, which is 144, 12, and 1,
that's my third row, and I'm going to subtract 144, 28.8,
and 5.76, and what am I going to get from there?
It will be 0, because this will cancel with this, -16.8, and -4.76,
that's what I'm going to get by doing the . . .
by doing the subtraction.
Now, remember this, this 5.76, the multiplier was used to make this to be 0,
which is the third row, first column become 0,
so this will correspond to third row, first column in the lower triangular matrix, the
value of 5.76.
So if you look at the end of first step . . . at the end of first step
of forward elimination, what I have is that I'll have a matrix which looks like this,
I will have 25, 5, 1, then I'll have a 0 here, -4.8 here, and -1.56 here, and then I'll have
a 0 here, -16.8 here, and -4.76 here.
So this has become 0, this has become 0 by doing the proper multiplication and
subtractions based on what we have in the first row.
And also what I've established is that the second row, first column of the lower
triangular matrix is 2.56, and the third row, first column is 5.76.
So those are the things which I have obtained
by doing this LU decomposition method here, or to
find the LU decomposition.
But I'm not done here, because I need to reduce this to the . . .
to an upper triangular matrix, do all the steps of forward elimination.So let's go ahead
and do that.
So now what I will have is that I'll have -4.8
right here, and I'll use that to make this to be 0.
So I'll have to divide the row two by -4.8 and multiply by 16.8 to
be able to find out . . . to make this to be 0.
So I'm going to write down the second row, which is 0,
-4.8, and -1.56, and I'm going to multiply it by -16.8, and I'm going
to divide by -4.8, and this is going to give me what?
It's going to give me 0, -16.8, -5.46,
that's what I'm going to get by multiplication, and now what I have to do is I have to subtract
this from the third row.
I'll have 0, -16.8, -4.76, and I'll have to subtract 0, -16.8, and
-5.46, and what I'm going to get is 0, 0, and 0.7.
So this -16.8 divided by -4.8 is nothing but 3.5.
So this number, 3.5, will be actually my third row, second column, because it
made this third row, second column to be 0, that's why that corresponds
to l32.
So if I write down now my . . . at the end of
second step, which is the last step of forward elimination, because
I always conduct . . . I always conduct n-minus-1 steps of forward elimination, I'll get what
follows, I'll get 25, 5, 1, first row stays the same, second row
stays the same, and the third row would be 0, 0,
0.7, and from here I'm getting l32 is equal to
3.5, okay?
Third row, second column of the lower triangular matrix
is 3.5.
So, having said this, this is, at the end of the forward
elimination steps, this is in fact your U matrix, which you are looking for in your
LU decomposition.
The upper triangular matrix which you get at the end of the forward elimination steps
is actually your U matrix, and how do establish your L matrix?
Your L matrix will be, which is a lower triangular matrix,
will be established by a 3-by-3, of course, and
you will have, of course, 0 here, 0 here, 0 here, because that's part of being a lower
triangular matrix.
You're going to put 1s in the diagonal . . . you're going to put 1s in the diagonal like this,
1, 1, 1.
And then in order to be able to fill this in and this in and this in, we have already
established what those are.
Like, for example, we said l21 is 2.56, because that is the
multiplier which made that term to be 0 in the U matrix, and this l31
is 5.76, and this l32, which we just found out, is
3.5.
So that's how you are able to establish what the lower triangular
elements will be in the lower triangular matrix.
So if I would write down what my LU decomposition is, I'll write it
as this, that, hey, this is my A, and this will be my L times U.
So A is 25, 5, 1, 64, 8, 1, 144, 12, 1, so this is my
original A matrix, and what is the L times U decomposition?
This will be my L, with 1s in the diagonal, 0s above the diagonal, and the
multipliers, corresponding multipliers at the proper places in the . . . below the
diagonal.
The U matrix is the same which I got at the end
of the forward elimination steps, so that's 25,
1, 0, -4.8, and -1.56, and 0, 0, and 0.7.
And you can verify by multiplying this matrix to this matrix, this
is my L matrix and this is my U matrix, and go ahead and use multiplication of matrix
operation to find out, to verify whether this L times U
gives you back this A matrix right here.
And that's how you do the LU decomposition of
a square matrix.
And that's the end of this segment.