Definition:Echelon Matrix/Reduced Echelon Form

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Definition

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix in echelon form whose order is $m \times n$.


The matrix $\mathbf A$ is in reduced echelon form if and only if, in addition to being in echelon form, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs.


Such a matrix is called a reduced echelon matrix.


Also known as

The reduced echelon form is also known as row canonical form, or reduced row echelon form.

The abbreviated term ref or rref is often used for reduced (row) echelon form, but it is recommended that it be explained when first invoked in an argument.

The definition of reduced column echelon form is directly analogous.


Examples

Arbitrary Example $1$

$\begin {bmatrix} 1 & 0 & -1 & 2 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix}$ is an echelon matrix, but not a reduced echelon matrix, because the leading $1$ in row $3$ is not the only $1$ in its column.


Arbitrary Example $2$

$\begin {bmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix}$ is a reduced echelon matrix.


Arbitrary Example $3$

$\begin {bmatrix} 0 & 1 & 0 & 2 \\ 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$ is not an echelon matrix, because the leading $1$ in row $2$ is to the left of the leading $1$ in row $1$.


Arbitrary Example $4$

$\begin {bmatrix} 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ \end {bmatrix}$ is not an echelon matrix, because row $2$ is a zero row, coming before row $3$.


Arbitrary Example $5$

$\begin {bmatrix} 1 & 5 & 4 & 2 \\ 0 & 6 & 0 & 9 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$ is not an echelon matrix, because the leading coefficient of row $2$ is not $1$.

It is, however, a non-unity variant of an echelon matrix.


Also see

  • Results about echelon matrices can be found here.


Linguistic Note

An echelon is:

a formation of troops, ships, aircraft, or vehicles in parallel rows with the end of each row projecting further than the one in front.

It derives from the French word échelon, which means a rung of a ladder, which describes the shape that this formation has when viewed from above or below.

It is pronounced e-shell-on or something like ay-shell-on, where the first ay is properly the French é.

Avoid the pronunciation et-chell-on, which is technically incorrect.


Sources

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