Reduced row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from 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

These two conditions imply that all entries in a column below a leading coefficient are zeros.

This is an example of a 3×5 matrix in row echelon form:

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:

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.

This is an example of a matrix in reduced row echelon form:

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 calculated using Euclidean division and without introducing any rational number or denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains non-integer coefficients.

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.

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