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 coefficient 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. These two conditions imply that all entries in a column below a leading coefficient are zeros. The following is an example of a 3×5 matrix in row echelon form, which is not in reduced row echelon form : Many properties of matrices may be easily deduced from their row echelon form, such as the rank and the kernel.
Reduced row echelon form
A matrix is in reduced row echelon form if it satisfies the following conditions:
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. This is an example of a matrix in reduced row echelon form, which shows that the left part of the matrix is not always an identity matrix: 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.
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; any matrix that is in echelon form can be put in an echelon form by adding a scalar multiple of a row to one of the above rows, for example: However, every matrix has a unique reduced row echelon form. In the above example, the reduced row echelon form can be found as 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.
