LU Decomposition Method: Decomposing a Matrix Example: Part 2 of 2

Uploaded by numericalmethodsguy on 09.03.2009

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.