Gaussian Elimination as a Matrix Factorization
Each of the elementary row operations from Gaussian Elimination (GE) has
associated with it a nonsingular matrix with the property that multiplying (on the
left) by that associated matrix gives the same result as applying the row operation.
1.
Row operation 1
(R1): Multiply row
i
by a scalar
α
6
= 0.
Associated matrix:
Let
D
be the identity matrix except for the (
i,i
)
element, which is
α
. Then the matrix
DA
is the result of R1 applied to
A
.
2.
Row operation 2
(R2): Interchange row
i
and row
j
.
Associated matrix:
Let
P
be the identity matrix with rows
i
and
j
interchanged. Then the matrix
PA
is the result of R2 applied to
A
.
3.
Row operation 3
(R3): Multiply row
j
by a scalar
m
and add it to row
i
.
Associated matrix:
Let
M
be the identity but with
m
replacing the 0 in the
(
i,j
) position. Then
M
=
I
+
me
i
e
t
j
, and the matrix
MA
is the result of R3
applied to
A
. Taking
m
=
m
ij
=

a
ij
/a
jj
puts a zero in the (
i,j
) position of
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 Fall '10
 MARK
 Linear Algebra, row operation, associated matrix

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