Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/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.
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.
- Results about echelon matrices can be found here.
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.