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In [[linear algebra]], a [[matrix (mathematics)|matrix]] is in '''echelon form''' if it has the shape resulting of a [[Gaussian elimination]]. '''Row echelon form''' means that Gaussian elimination has operated on the rows and
'''column echelon form''' means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its [[transpose]] is in row echelon form. Therefore only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.


Specifically, a matrix is in '''row echelon form''' if
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* All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix).
* The [[Leading coefficient#Linear algebra|leading coefficient]] (the first nonzero number from the left, also called the [[pivot element|pivot]]) of a nonzero row is always strictly to the right of the leading coefficient of the row above it (some texts add the condition that the leading coefficient must be 1.<ref>See, for instance, {{harvtxt|Leon|2009|p=13}}</ref>).
* All entries in a column below a leading entry are zeroes (implied by the first two criteria).<ref>{{harvnb|Meyer|2000|p=44}}</ref>
 
This is an example of a 3×5 matrix in row echelon form:
 
<math>
\left[ \begin{array}{ccccc}
1 & a_0 & a_1 & a_2 & a_3 \\
0 & 0 & 2 & a_4 & a_5 \\
0 & 0 & 0 & 1 & a_6
\end{array} \right]
</math>
 
=={{anchor|rref}}Reduced row echelon form==
A matrix is in '''reduced row echelon form''' (also called '''row canonical form''') if it satisfies the following conditions:
* It is in row echelon form.
* Every leading coefficient is 1 and is the only nonzero entry in its column.<ref>{{harvnb|Meyer|2000|p=48}}</ref>
 
The reduced row echelon form of a matrix may be computed by [[Gauss–Jordan elimination]]. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it.
 
This is an example of a matrix in reduced row echelon form:
 
<math>
\left[ \begin{array}{ccccc}
1 & 0 & 0 & 0 & b_1 \\
0 & 1 & 0 & 0 & b_2 \\
0 & 0 & 0 & 1 & b_3
\end{array} \right]
</math>
 
Note that this does not always mean that the left of the matrix will be an [[identity matrix]], as this example shows.  
 
For matrices with integer coefficients, the [[Hermite normal form]] is a row echelon form that may be computed using [[Euclidean division]] and without introducing any [[rational number]] nor denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains non-integer entries.
 
== Transformation to row echelon form ==
 
By means of a finite sequence of [[elementary row operations]], called [[Gaussian elimination]], any matrix can be transformed to row echelon form.  Since elementary row operations preserve the [[row space]] of the matrix, the row space of the row echelon form is the same as that of the original matrix.
 
The resulting echelon form is not unique; for example, any multiple by a scalar of a matrix in echelon form is also an echelon form of the same matrix. However, every matrix has a unique ''reduced'' row echelon form. This means that the nonzero rows of the reduced row echelon form are the unique reduced row echelon generating set for the row space of the original matrix.
 
== Systems of linear equations ==
 
A [[system of linear equations]] is said to be in ''row echelon form'' if its [[augmented matrix]] is in row echelon form. Similarly, a system of equations is said to be in ''reduced row echelon form'' or ''canonical form'' if its augmented matrix is in reduced row echelon form.
 
The canonical form may be viewed as an explicit solution of the linear system. In fact, the system is [[System of linear equations#Consistency|inconsistent]], if and only if one of the equations of the canonical form is reduced to 1 = 0. Otherwise, regrouping in the right hand side all the terms of the equations, but the leading ones expresses the variables corresponding to the pivots as constants or linear functions of the other variables, if any.
 
==Notes==
{{reflist}}
 
==References==
* {{Citation | last1 = Leon | first1 = Steve | title = Linear Algebra with Applications | isbn=978-0136009290 | edition = 8th | year = 2009 | publisher = Pearson }}.
* {{Citation | last1=Meyer | first1=Carl D. | title=Matrix Analysis and Applied Linear Algebra | url=http://www.matrixanalysis.com/ | publisher=[[Society for Industrial and Applied Mathematics|SIAM]] | isbn=978-0-89871-454-8 | year=2000}}.
 
==External links==
{{wikibooks|Linear Algebra|Row Reduction and Echelon Forms}}
 
*[http://people.revoledu.com/kardi/tutorial/LinearAlgebra/RREF.html Interactive Row Echelon Form with rational output]
 
{{Numerical linear algebra}}
 
[[Category:Numerical linear algebra]]
 
[[de:Lineares Gleichungssystem#Stufenform, Treppenform]]

Revision as of 16:52, 19 February 2014

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