Mathematics-Online lexicon: Echelon Form

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	Mathematics-Online lexicon:
	Echelon Form

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

overview

By Gaussian elimination any LSE can be brought to row-echelon form:

$\displaystyle Ax = b \rightarrow \underbrace{ \left(\begin{array}{cccc ccc} ... ...ht) = \left(\begin{array}{c} c_1 \\ \vdots \\ c_m \end{array}\right)\, ,$

where the so called pivots

$\displaystyle p_1=a'_{1,j_1},\ldots,p_k=a'_{k,j_k},\quad 1\le j_1<\cdots<j_k\le n\, ,$

are not equal to zero and

is the rank of

In detail the $\ell$ -th elimination step proceeds as follows:

Select as pivot $p_\ell$ an element $\ne 0$ with smallest possible column-index $j_\ell$ and row-index $\ge \ell$ . Interchange the $\ell$ -th equation with the equation which contains the pivot.
If there is no possible pivot, the echelon form is reached.
For $i>\ell$ an appropriate multiple of the $\ell$ -th equation is subtracted from the -th equation so that $a_{i,j_\ell}$ equals zero.

(Authors: Burkhardt/Höllig/Streit)

Example:

Scheme of the Flow

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automatically generated 3/16/2005