# Where did the German term “Spur” of a matrix come from?

I wonder the origin of the term "trace" of a matrix.

As I googled, it was the English translation of the German word "Spur" and it appeared in the translation of H. Weyl's Raum, Zeit, Materie. http://jeff560.tripod.com/t.html

Recently, I found an article to mention the origin of "Spur". http://senseis.xmp.net/?JapaneseGoTerms%2FDiscussion

I quoted the paragraph from the article.

ilan: Remind me of G.H. Hardy using the term "quadratfrei" because he said he couldn't find a good English equivalent. Looking back on it years later, I suppose it was a joke. P.J. Cohen tells the story that the use of "trace" in matrix theory comes from the translation of the German word "spur" which means trace, but which was used by Germans who simply took the English name "spur" given by Cayley because the main diagonal looks like a spur. I have never checked its apocryphality (not a real word). Here is something I do know about: The English term "continued fraction" should be "fraction continué" in French but has been corrupted in the last 100 years into "fraction continue." Recently, some famous mathematicians working at Orsay have translated this corruption into English publishing a paper on "continuous fractions" despite all their English references usage of the correct term. One can wonder at their lack of scholarship, or whether it is an elaborate joke. In any case, everyone I ever mentioned this to didn't care except for wondering why I did.

I think it's very interesting story. Is it true?

If it's not a true story, why on earth did Germans call it "Spur"?

• They say the German word Spur is the English word spoor, which refers to the trail or track left by an animal; if you are hunting a wild animal, you follow its spoor. This makes more sense to me, the main diagonal of a matrix thought of as a trail through the jungle or grasslands of, in particular, South Africa. en.wiktionary.org/wiki/spoor – Will Jagy May 20 '15 at 23:21
• Where did the German term “Spur” of a matrix come from? - Keine Ahnung ! Ich bin... spurlos ! :-$)$ – Lucian May 20 '15 at 23:27
• @ZubinMukerjee Thank you for your correction. English is not easy to non-English speakers. :-) – P.-S. Park May 20 '15 at 23:29
• "The term trace is a calque from the German Spur (cognate with the English spoor), which, as a function in mathematics, is often abbreviated to "tr"." en.wikipedia.org/wiki/Trace_%28linear_algebra%29 and en.wiktionary.org/wiki/Spur#German – Will Jagy May 20 '15 at 23:41
• For what it’s worth, a search of the University of Michigan Historical Mathematics Collection turns up only two instance of spur in Cayley’s collected works, and they have to do with physical gears, not matrices. – Brian M. Scott May 24 '15 at 10:18