User:Jim.belk/Draft:Row echelon form

In linear algebra, a matrix is in row echelon form (or echelon form) if it satisfies the following requirements:


 * 1) Any rows of zeros are at the bottom of the matrix.
 * 2) In each of the remaining rows, the first nonzero entry is a one.  (These entries are called pivots.)
 * 3) The pivot in each row lies to the right of the pivot in the preceding row.

Visually, a matrix in echelon form has a "staircase" of zeros in the lower left:
 * $$\begin{pmatrix}

1 & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast \\ 0 &  1  & \ast & \ast & \ast & \ast & \ast & \ast & \ast \\ 0 &  0  &   0  &   1  & \ast & \ast & \ast & \ast & \ast \\ 0 &  0  &   0  &   0  &   0  &   0  &   1  & \ast & \ast \\ 0 &  0  &   0  &   0  &   0  &   0  &   0  &   1  & \ast \\ 0 &  0  &   0  &   0  &   0  &   0  &   0  &   0  &   0  \\ 0 &   0  &   0  &   0  &   0  &   0  &   0  &   0  &   0 \end{pmatrix}$$

The first nonzero entry in each row (circled above) is called a pivot.

Any matrix can be put into echelon form through Gaussian elimination. (In some books, Gaussian elimination involves changing each pivot to a one, and this is taken as a requirement for echelon form.) The echelon form makes various properties of the original matrix apparent, including the rank and determinant of the matrix, the dimension of the null space, the linear dependence relations between the columns, and so forth.

Systems of linear equations
Row echelon form is closely related to reduced row echelon form (row canonical form). The difference is that in reduced row echelon form, the entries above the leading coefficient also have to be zero, and that the leading coefficient is always one.

The first non-zero entry in each row is called a pivot.

Examples
This matrix is in row echelon form:



\begin{bmatrix} 0 & 1 & 4 & 0 & -3 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1  \\ 0 & 0 & 0 & 0 & 0  \\ \end{bmatrix} $$

The following matrix is also in row echelon form:



\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 9 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ \end{bmatrix} $$

However, this matrix is not in row echelon form, as the leading coefficient of row 3 is not strictly to the right of the leading coefficient of row 2.



\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 3 & 7 & 2 \\ 0 & 2 & 0 & 0 \\ \end{bmatrix} $$

Non-uniqueness
Every non-zero matrix can be reduced to an infinite number of echelon forms (they can all be multiples of each other, for example) via elementary matrix transformations. However, all matrices and their row echelon forms correspond to exactly one matrix in reduced row echelon form.

Systems of linear equations
A system of linear equations is said to be in echelon form if its augmented matrix is in row echelon form. Similarly, a system of equations is said to be in reduced echelon form or canonical form if its augmeneted matrix is in reduced row echelon form.