# Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1

## Definition

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

$\mathbf A$ is in **non-unity echelon form** if and only if:

- $(1): \quad$ Each row (except perhaps row $1$) starts with a sequence of zeroes
- $(2): \quad$ Except when for row $k$ and row $k + 1$ are zero rows, the number of zeroes in this initial sequence in row $k + 1$ is strictly greater than the number of zeroes in this initial sequence in row $k$
- $(3): \quad$ The non-zero rows appear before any zero rows.

## Also known as

An **Echelon Matrix** and a matrix in **echelon form** are the same thing.

A matrix in **echelon form** is also sometimes seen as being in **row echelon form**.

It is noted that there appears to be no equivalent definition in the literature for the concept of **column echelon form**, although its structure would be analogous.

## Also defined as

Many sources require that, for a matrix to be in **echelon form**, the leading coefficient in each non-zero row must be $1$.

Such a matrix is detailed in 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

**, where the first**

*ay*-shell-on**ay**is properly the French

**é**.

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

## Sources

- 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations