# What does echelon mean?

When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?

-
I assume it comes from the military term for a structured arrangement of troops into parallel lines. – Bye_World Jan 26 at 20:14
@Bye_World: Please write this as an answer (with maybe a link to something) so that I can upvote and accept. – John Doe Jan 26 at 20:16
I'm not going to write up an answer because this is just a guess. You can research the etymology of the word yourself if you're really interested.... But here's a cool picture from the Wikipedia page on Echelon formation that's pretty suggestive. – Bye_World Jan 26 at 20:17
I thought it was a cycling term to describe how the peleton spreads out. This is akin to having the reduced matrix. – Karl Jan 27 at 17:48

echelon (n.)

1796, echellon, "step-like arrangement of troops," from French échelon "level, echelon," literally "rung of a ladder," from Old French eschelon, from eschiele "ladder," from Late Latin scala "stair, slope," from Latin scalae (plural) "ladder, steps," from PIE $\ast$skand- "to spring, leap" (see scan (v.)). Sense of "level, subdivision" is from World War I.

Steven Schwartzman's The Words of Mathematics: An Etymological Dictionary of Mathematical Terms (MAA, 1994) also attributes the root of the term to the French word échelon and its meaning of "rung of a ladder" (link from Google Books).

The earliest appearance of "echelon form" as a linear algebra term that I can find is in Birkhoff and Mac Lane (1953), A Survey of Modern Algebra (link from Google Books). Yet, Nathan Jacobson's Lectures in Abstract Algebra: II. Linear Algebra (1953) does not contain this term. I guess the term was introduced to linear algebra more or less around the 1950s.

-
A real success story. Birkhoff and MacLane introduce the terminology, and now all textbooks use it. With the use of computers, how long will it be until everyone forgets it? – GEdgar Jan 27 at 14:54

As has already been suggested, it has its origins in military vocabulary.

The Wiki article on echelon formation contains many pictures of stuff in echelon formation, and you can immediately see why one might say that the rows of a row-echelon matrix are in "echelon formation." I would have liked to include the wiki images, but for some reason they would not load. I found another one instead:

-
This is a nice explanation but I doubt the origin of the terminology is the military term. Instead, I'd assume both the military and the mathematical usage derive from the fact that echelon means (or at least meant, not sure) something like steps (of a ladder). – quid Jan 26 at 23:56
Dear @quid : hrm, it seems by far to be the most likely origin of the mathematical usage to me. Yes, to be pedantic, echelon derives from "rung," but "echelon form" evokes the military usage far more. Regards – rschwieb Jan 27 at 2:30
It could be interesting to investigate this historically. My reasoning is based mainly on how it is called in other languages. If you change on en.wikipedia.org/wiki/Row_echelon_form to other languages several seem to employ terms that are seem compatible much more with steps or staircase idea than with a military formation. – quid Jan 27 at 7:52
In Swedish it is "trappstegsmatris" and searching for "trappsteg" I get images of steps (more staircases than ladders though). German it says "stufenform" or "treppenform" in line with steps and staircases. For the Polish "schodkowa" one finds the staircase-like pyramids when searching. The word is also used in what is called "step function" in English. – quid Jan 27 at 7:52
@quid interesting, but I don't buy the connection between rung and stair step. They have decidedly different shapes, and staircases are more like an echelon formation than a ladder. They could have all derived from "echelon formation" and "stair step" directly because of their shapes. – rschwieb Jan 27 at 9:31

It simply means the rows are ordered in a unique hierarchy by the positions of their leading ones (rows of all zeros being lumped at the bottom).

-
@Arthur: Is $\begin{pmatrix}1 & 0 &0 \\ 0&0& 1\end{pmatrix}$ in row echelon form? – Roland Jan 26 at 20:19
@Arthur: Apparently you are thinking of the positions of leading ones but even these are not necessarily diagonal. The word echelon here is applied to the rows and has the sense of level or rank within the matrix. – hardmath Jan 26 at 20:19
@Roland Yes, it is. – immibis Jan 26 at 23:39