CHAPTER 04.07: LU DECOMPOSITION: Decomposing a Square Matrix: Part 2 of 2
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. |