A First Course in Linear Algebra, 2017a

2.2. LU Factorization 101
Thenextstepistotake2timesthetoprowandaddtothe bottom in the matrix on the right. To ensure that
the product is unchanged, you place a −2 in the bottom left in the matrix on the left. Thus the next step
yields
⎡
⎣
1 0 0
2 1 0
−2 0 1
⎤⎡
⎦⎣
1 2 3
0 −1 −5
0 7 4
Next take 7 times the middle row on right and add to bottom row. Updating the matrix on the left in a
similar manner to what was done earlier,
⎡
1 0
⎤⎡
0 1 2
⎤
3
⎣ 2 1 0 ⎦⎣
0 −1 −5 ⎦
−2 −7 1 0 0 −31
⎤
⎦
At this point, stop. You are done.
♠
The method just described is called the multiplier method.
2.2.3 Solving Systems using LU Factorization
One reason people care about the LU factorization is it allows the quick solution of systems of equations.
Here is an example.
Example 2.69: LU factorization to Solve Equations
Suppose you want to find the solutions to
⎡
⎣
1 2 3 2
4 3 1 1
1 2 3 0
⎡
⎤
⎦⎢
⎣
x
y
z
w
⎤
⎡
⎥
⎦ = ⎣
1
2
3
⎤
⎦.
Solution.
Of course one way is to write the augmented matrix and grind away. However, this involves more row
operations than the computation of the LU factorization and it turns out that the LU factorization can give
the solution quickly. Here is how. The following is an LU factorization for the matrix.
⎡
⎣
1 2 3 2
4 3 1 1
1 2 3 0
⎤
⎡
⎦ = ⎣
1 0 0
4 1 0
1 0 1
⎤⎡
⎦⎣
1 2 3 2
0 −5 −11 −7
0 0 0 −2
Let UX = Y and consider LY = B where in this case, B =[1,2,3] T . Thus
⎡ ⎤⎡
⎤ ⎡ ⎤
1 0 0 y 1 1
⎣ 4 1 0 ⎦⎣
y 2
⎦ = ⎣ 2 ⎦
1 0 1 y 3 3
⎤
⎦.