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Concept
Row echelon form
Summary
In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.
A matrix being in 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
All rows consisting of only zeroes are at the bottom.
The leading entry (that is the left-most nonzero entry) of every nonzero row is to the right of the leading entry of every row above.
Some texts add the condition that the leading coefficient must be 1 while others regard this as reduced row echelon form.
These two conditions imply that all entries in a column below a leading coefficient are zeros.
The following is an example of a 4x5 matrix in row echelon form, which is not in reduced row echelon form (see below):
Many properties of matrices may be easily deduced from their row echelon form, such as the rank and the kernel.
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.
The leading entry in each nonzero row is a 1 (called a leading 1).
Each column containing a leading 1 has zeros in all its other entries.
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. For a given matrix, despite the row echelon form not being unique, all row echelon forms and the reduced row echelon form have the same number of zero rows and the pivots are located in the same indices.
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