Lecture 4
Gaussian Elimination
The basic idea behind the Gaussian elimination method is to transform general linear
system of equations
b
Ax
=
into triangular form. To do this we need to replace selected
nonzero entries of matrix by zeros.
This can be accomplished by taking linear
combinations of rows.
Let us first see how Gaussian elimination works. Consider the following system of
equations
0
1
2
2
6
3
2
1
3
2
1
3
2
1
=
−
+
=
+
−
=
+
+
x
x
x
x
x
x
x
x
x
The matrix representation of this system is
=
−
−
⇒
=
0
1
6
1
1
1
1
2
2
1
1
1
3
2
1
x
x
x
b
Ax
Now there are several operations that one can perform on a system of equations, without
changing its solution.
1.
We can replace any equation by a nonzero constant times the original equation
2.
We can replace any equation by its sum with another equation
3.
We can carry out the above operations any number of times.
In the matrix language the operations we apply on augmented matrix
[ ][ ]
−
−
=
0
1
6
1
1
1
1
2
2
1
1
1
b
A
Thus for example we can add –2 times the first row to the second row and –1 times the
first row to the third row to obtain the following equivalent system
−
−
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 Spring '09
 EricdeSturler
 Gaussian Elimination, LU factorization, 2 L

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