§1 - National Tsing Hua University
§3.3 Gaussian elimination
Consider the following example
Example 3.1:
[pic]
([pic](, [pic]
([pic](, [pic]
[pic]
([pic](, [pic], yieds
[pic]
Here [pic], [pic], [pic] are called multipliers.
Then we solve [pic], [pic], [pic] by backward substitution [pic], [pic], [pic]. Use the matrix notation, we have
[pic]
From Example 3.1 we see that we reduce the problem of [pic] by eliminations into [pic] where [pic] is a upper triangular matrix.
[pic]
where we assume [pic], [pic], [pic]are called pivots. [pic] becomes
[pic]
We solve it by backward substitution,
[pic]
[pic], [pic].
In fact Gaussian elimination is equivalent to [pic] decomposition of matrix [pic]
[pic]
where
[pic] [pic]
where [pic] are multipliers. From Example 3.1
we have
[pic]
Remark 3.1: For the linear system of [pic] equations in [pic] unknowns i.e. [pic], [pic], [pic], [pic], we can apply elimination process to obtain a kind of “staircase pattern” or echelon form
[pic][pic] [pic]
and
[pic], [pic] is a [pic] lower triangular matrix of the form [pic]
Remark 3.2: Finding [pic] by Gaussian elimination. Let [pic] and the identity matrix [pic] can be written as [pic] where [pic].
Then [pic] can be written as
[pic]
or
[pic], [pic] [pic]
We say solve [pic] by Gaussian eliminations
[pic] [pic]
If we do more elimination process to reduce [pic] to [pic] (This is called Gauss-Jordan elimination) Then
[pic].
Example 3.2: Find [pic] where [pic] is given in Example 3.1
[pic]
[pic]
[pic]
[pic]
Remark 3.3: There is a analytic formula for [pic], [pic] where [pic]
[pic] [pic]
where [pic] is the [pic] submatrix of [pic] obtained by deleting [pic]-th row and [pic]-th column.
Remark 3.4: If we solve [pic] by Gaussian elimination, it costs [pic] multiplications. It cost to [pic] compute [pic] by Gauss-Jordan method. Hence in numerical computation, we don’t solve [pic] by [pic]. But in MATLAB, they solve it this way.
For the analytic formula [pic] of [pic], we never use it in numerical computation. If we compute [pic] directly by definition of [pic], it will cost [pic] multiplication. In fact if you want to compute [pic], you may use Gaussian elimination. For [pic], [pic].
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