Row Echelon Form

In linear algebra, a 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

  • 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 (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.
  • All entries in a column below a leading entry are zeroes (implied by the first two criteria).

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

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

A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination. This means that it satisfies the additional condition:

  • Every leading coefficient is 1 and is the only nonzero entry in its column, as in this example:

\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]

Note that this does not always mean that the left of the matrix will be an identity matrix. For example, the following matrix is also in reduced row-echelon form:

\left[ \begin{array}{cccccc} 1 & 0 & 0 & 1/2 & 0 & b_1 \\ 0 & 0 & 1 & -1/3 & 0 & b_2 \\ 0 & 0 & 0 & 0 & 1 & b_3 \end{array} \right]

Read more about Row Echelon Form:  Transformation To Row Echelon Form, Systems of Linear Equations

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