Linear Algebra, Theory And Applications, 2012a

5.5. JUSTIFICATION FOR THE MULTIPLIER METHOD 127
Let Ux = y and consider PLy = b. In other words, solve,
⎛
⎝ 1 0 0
⎞ ⎛
0 0 1 ⎠ ⎝ 1 0 0
⎞ ⎛
4 1 0 ⎠ ⎝ y ⎞ ⎛
1
y 2
⎠ = ⎝ 1 2
0 1 0 1 0 1 y 3 3
Then multiplying both sides by P gives
⎛
⎝ 1 0 0
4 1 0
1 0 1
⎞ ⎛ ⎞ ⎛ ⎞
⎠ ⎝ y 1
y 2
y 3
⎠ = ⎝ 1 3 ⎠
2
⎞
⎠ .
and so
⎛
y = ⎝
y 1
y 2
y 3
⎞ ⎛
⎠ = ⎝
Now Ux = y and so it only remains to solve
⎛
⎛
⎝ 1 2 3 2 ⎞
0 −5 −11 −7 ⎠ ⎜
⎝
0 0 0 −2
1
−1
1
x 1
x 2
x 3
x 4
⎞
⎠ .
⎞
⎛
⎟
⎠ = ⎝
1 −1
1
⎞
⎠
which yields
⎛
⎜
⎝
x 1
x 2
x 3
x 4
⎞ ⎛
⎟
⎠ = ⎜
⎝
1
5 + 7 5 t
9
10 − 11
5 t
t
− 1 2
⎞
⎟
⎠ : t ∈ R.
5.5 Justification For The Multiplier Method
Why does the multiplier method work for finding an LU factorization? Suppose A is a
matrix which has the property that the row reduced echelon form for A may be achieved
using only the row operations which involve replacing a row with itself added to a multiple
of another row. It is not ever necessary to switch rows. Thus every row which is replaced
using this row operation in obtaining the echelon form may be modified by using a row
which is above it. Furthermore, in the multiplier method for finding the LU factorization,
we zero out the elements below the pivot entry in first column and then the next and so on
when scanning from the left. In terms of elementary matrices, this means the row operations
used to reduce A to upper triangular form correspond to multiplication on the left by lower
triangular matrices having all ones down the main diagonal and the sequence of elementary
matrices which row reduces A has the property that in scanning the list of elementary
matrices from the right to the left, this list consists of several matrices which involve only
changes from the identity in the first column, then several which involve only changes from
the identity in the second column and so forth. More precisely, E p ···E 1 A = U where U
is upper triangular, each E i is a lower triangular elementary matrix having all ones down
the main diagonal, for some r i , each of E r1 ···E 1 differs from the identity only in the first
column, each of E r2 ···E r1 +1 differs from the identity only in the second column and so
Will be L
{ }} {
forth. Therefore, A = E1 −1 ···Ep−1 −1 E−1 p U. You multiply the inverses in the reverse order.
Now each of the E −1
i is also lower triangular with 1 down the main diagonal. Therefore
their product has this property. Recall also that if E i equals the identity matrix except