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

• 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), and
• the leading coefficient (the first nonzero number from the left, also called the 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^[1]).

These two conditions imply that all entries in a column below a leading coefficient are zeros. ^[2]

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

$\left[ \begin{array}{ccccc} 1 & a_0 & a_1 & a_2 & a_3 \\ 0 & 0 & 2 & a_4 & a_5 \\ 0 & 0 & 0 & 1 & a_6 \end{array} \right]$

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 (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.^[3]

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.^[4] 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.^[4]

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

$\left[{\begin{array}{ccccc}1&0&a_{1}&0&b_{1}\\0&1&a_{2}&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]$

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.

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 (equivalent) echelon form by adding a scalar multiple of a row to one of the above rows, for example:

$\begin{bmatrix} 1 & 3 & -1 \\ 0 & 1 & 7 \\ \end{bmatrix} \xrightarrow{\text{add row 2 to row 1}} \begin{bmatrix} 1 & 4 & 6 \\ 0 & 1 & 7 \\ \end{bmatrix}.$

However, every matrix has a unique reduced row echelon form. In the above example, the reduced row echelon form can be found as

$\begin{bmatrix} 1 & 3 & -1 \\ 0 & 1 & 7 \\ \end{bmatrix} \xrightarrow{\text{subtract 3 times row 2 from row 1}} \begin{bmatrix} 1 & 0 & -22 \\ 0 & 1 & 7 \\ \end{bmatrix}.$

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 in 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 inconsistent, if and only if one of the equations of the canonical form is reduced to 0 = 1.^[5] 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

1. See, for instance, Leon (2009, p. 13)
2. Meyer 2000, p. 44
3. Meyer 2000, p. 48
4. Anton, Howard; Rorres, Chris (2013-10-23). Elementary Linear Algebra: Applications Version, 11th Edition. Wiley Global Education. p. 21. ISBN 9781118879160.
5. Cheney, Ward; Kincaid, David R. (2010-12-29). Linear Algebra: Theory and Applications. Jones & Bartlett Publishers. pp. 47–50. ISBN 9781449613525.

References

• Leon, Steve (2009), Linear Algebra with Applications (8th ed.), Pearson, ISBN 978-0136009290.
• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8.

External links

• Interactive Row Echelon Form with rational output