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

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Definition

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


Definition 1

$\mathbf A$ 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.


Definition 2

$\mathbf A$ is in non-unity echelon form if and only if it contains no adjacent rows of the form:

$\begin {pmatrix} 0 & 0 & \cdots & 0 & x_1 & x_2 & \cdots \\ 0 & 0 & \cdots & 0 & y_1 & y_2 & \cdots \\ \end {pmatrix}$

where:

$(1): \quad y_1 \ne 0$
$(2): \quad x_1$ can be any value at all, including $0$.


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

Some sources do not 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 Non-Unity Variant of Echelon Matrix.


Examples

Arbitrary Matrix 1

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


Arbitrary Matrix $2$

$\begin {bmatrix} 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ \end {bmatrix} $ is not an echelon matrix, because the leading coefficient of row $2$ is in the same column as that of the row above it.


Arbitrary Matrix $3$

$\begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 2 & 2 & 2 \\ 0 & 0 & 3 & 4 \\ \end {bmatrix} $ is a non-unity echelon matrix.


Arbitrary Matrix $4$

$\begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ \end {bmatrix} $ is not an echelon matrix, because the leading coefficient of row $3$ is in the same column as that of the row above it.


Arbitrary Matrix $5$

$\begin {bmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 2 & 2 & 2 \\ 0 & 0 & 0 & 3 & 3 \\ \end {bmatrix} $ is a non-unity 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.