echelon

Etymology

under the pedestal of the former National Kaiser Wilhelm Monument in Berlin, Germany. The word echelon is derived from French and Latin words meaning “ladder”.]] Borrowed from French échelon (“rung; echelon”), from échelle (“ladder”) + -on (“suffix forming diminutives”). Échelle is derived from Latin scāla (“ladder”), from scandō (“to ascend, climb”), from Proto-Indo-European *skend- (“to jump”).

noun

1. A level or rank in an organization, profession, or society.
   Other important functions performed by the GP [general practitioner] are those of referring patients to other (health) care facilities and acting as contact person for other providers of aid, both for other facilities in first echelon care and with respect to second echelon care (outpatient care and treatment in hospital). 1987, “Scenario A: The Reference Scenario”, in C. F. Hollander, H. A. Becker, editors, Growing Old in the Future: Scenarios on Health and Ageing 1984–2000: Scenario-report, Commissioned by the Steering Committee on Future Health Scenarios, Dordrecht: Martinus Nijhoff Publishers, →DOI, section 2.5.3 (GP Care), page 85
   Officials in China are divided into three echelons, or generations. The first echelon joined the Communist Party in the 1920s, soon after the party was founded. […] The second echelon joined the Communist Party before the founding of the People's Republic of China in 1949. The party was then still struggling, and many of the second echelon died during the wars. They were again true patriots. The third echelon joined the Communist Party after the founding of the People's Republic. 2004, Alfred Kuo-liang Ho, “Deng’s Political Reforms”, in China’s Reforms and Reformers, Westport, Conn.: Praeger Publishers, Greenwood Publishing Group, page 120
2. (cycling) A line of riders seeking maximum drafting in a crosswind, resulting in a diagonal line across the road.
   In an echelon, in which several cyclists are sitting in on one another, each rider takes his turn of about 200 meters at the front before dropping to the rear. 1972, Journal of the American Association for Health, Physical Education, and Recreation, volume 43, [Washington, D.C.]: American Association for Health, Physical Education, and Recreation, →OCLC, page 22, column 2
   Cyclists in an echelon take up more room in the traffic lane than they do when riding single or double file. 1989, Fred Matheny, Bicycling Magazine’s Complete Guide to Riding and Racing Techniques, Emmaus, Pa.: Rodale Press, page 99
3. (military) A formation of troops, ships, etc., in diagonal parallel rows.
   The troops selected by his Royal Highness for this daring exploit, consisted of the war battalions of the 3d, 6th, and 7th regiments of the Royal Guard, forming the first echelon, […] 1833 October, “a royalist”, “Sketches of the War of the French in Spain in the Year 1823”, in The United Service Journal and Naval and Military Magazine, part III, number 59, London: Published for Henry Colburn, by Richard Bentley, New Burlington Street, →OCLC, page 183
   The order in echelons is favorable for attack, because it readily conforms to the nature of the ground, and does not necessitate engaging more than a part of the forces; it is adopted for the purpose of attacking a particular point of the enemy's line. 1863, Ed. [Édouard] De La Barre Duparcq, “Infantry Formation and Tactics”, in George W[ashington] Cullum, transl., Elements of Military Art and History: Comprising the History and Tactics of the Separate Arms; the Combination of the Arms; and the Minor Operations of War, New York, N.Y.: D[avid] Van Nostrand, 192 Broadway, page 84
   As soon as the infantry had replenished their ammunition, they wheeled to the left in échelon of brigades, and began to march towards Surgham ridge. The movements of a great force are slow. It was not desirable that the British division, which led the échelon, should remain in the low ground north of Surgham—where it was commanded, had no field of fire, and could see nothing—and accordingly both these brigades moved forward almost together to occupy the crest of the ridge. 1899, Winston Spencer Churchill, “The Battle of Omdurman: September 2, 1898”, in F[rancis] Rhodes, editor, The River War: An Historical Account of the Reconquest of the Soudan … In Two Volumes, volume II, London: Longmans, Green, and Co., 89 Paternoster Row, →OCLC, pages 145–146
   “The first echelon of the Russian force was stuck without heat, without oil, without bombs and without gas. And it all happened because of the work of 30 people,” Honchar said. 2022-03-28, Julian Borger, quoting Yaroslav Honchar, “The drone operators who halted Russian convoy headed for Kyiv”, in The Guardian

verb

1. (transitive, military) To form troops into an echelon.
   July 1, Montcalm made a movement in advance, echeloning his troops from Fort Carillon to the foot of Lake George, to curb the enemy, and obstruct their landing. 1860, François-Xavier Garneau, “Battle of Carillon (Ticonderoga). 1758.”, in Andrew Bell, transl., History of Canada, from the Time of Its Discovery till the Union Year (1840–1): Translated from “L’Histoire du Canada” of F. X. Garneau, Esq., and Accompanied with Illustrative Notes, etc., etc. … In Three Volumes, volume II, Montreal, Que.: Printed and published by John Lovell, St. Nicholas Street, →OCLC, page 205
   Behind the 17th Panzer Division the corps echeloned the main force of its other division, the 16th Panzer Division. 1968, Earl F[rederick] Ziemke, “Offensives on Both Flanks—the South Flank”, in Stalingrad to Berlin: The German Defeat in the East (Army Historical Series), Washington, D.C.: Office of the Chief of Military History, U.S. Army, →OCLC; republished Washington, D.C.: Center of Military History, United States Army, 2002, →OCLC, page 225

adj

1. (linear algebra) Of a matrix: having undergone Gaussian elimination with the result that the leading coefficient or pivot (that is, the first nonzero number from the left) of a nonzero row is to the right of the pivot of the row above it, giving rise to a stepped appearance in the matrix.
   An echelon matrix is a matrix, not necessarily square, with the following two properties: (i) There is at least one non-zero entry; rows consisting entirely of zeros, if any, lie below rows with at least one non-zero entry. (ii) In each non-zero row after the first, the left-most non-zero entry lies to the right of the left-most non-zero entry in the preceding row. […] In each of the non-zero rows of an echelon matrix, the left-most non-zero entry is called the pivot, […] 2001, Malcolm Pemberton, Nicholas Rau, “Systems of Linear Equations”, in Mathematics for Economists: An Introductory Textbook, Manchester, New York, N.Y.: Manchester University Press, section 12.1 (Echelon Matrices), page 204