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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"?

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    $\begingroup$ 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 $\endgroup$
    – Will Jagy
    May 20, 2015 at 23:21
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    $\begingroup$ Where did the German term “Spur” of a matrix come from? - Keine Ahnung ! Ich bin... spurlos ! :-$)$ $\endgroup$
    – Lucian
    May 20, 2015 at 23:27
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    $\begingroup$ "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 $\endgroup$
    – Will Jagy
    May 20, 2015 at 23:41
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    $\begingroup$ 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. $\endgroup$ May 24, 2015 at 10:18
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    $\begingroup$ Just for the record, Issai Schur uses the concept of Spur on page 6 of his dissertation (1901, gdz.sub.uni-goettingen.de/id/PPN271034092) and seems to assume his audience is acquainted with the concept. Since his main audience was Frobenius, I'd look further in his works. $\endgroup$ Jun 26, 2022 at 7:29

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For all we know, Dedekind introduced what in modern terminology is the field trace as "Spur" on page 5 of his article Über die Discriminanten endlicher Körper. (In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 1882), as a third rational invariant for algebraic numbers, after the already known discriminants and norms in algebraic field extensions.

Dedekind did notice that this Spur can be computed via summing the diagonal elements of a matrix, but did not make much use of that; he mostly viewed it as a sum of Galois conjugates, and discussed it at length in his Supplement XI to the fourth edition (1894) of Dirichlet's Vorlesungen über Zahlentheorie. By 1897, this use had been adopted by an algebra textbook of Weber's and in an article of Hensel's.

When Frobenius founded character theory in 1896, he did so without using matrices at all; when he brought matrices into play in 1897, he noticed that his characters come out as sums of diagonal elements of matrices, but did not give that a special name; only in his 1899 report Über die Composition der Charaktere einer Gruppe he writes (bottom of first page):

Nennt man nach dem Vorgange von DEDEKIND die Summe der Diagonalelemente einer Substitution oder Matrix ihre Spur, [...]

that is, he ("following Dedekind") takes the sum-of-diagonal definition for any matrix, not just ones coming from field elements, and might have been the first to do so. Frobenius' doctoral student Schur followed suit on page 6 of his (1901) thesis Ueber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, and certainly it spread widely after this.

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D.E. Littlewood used the term in "the theory of group characters and matrix representations of groups" circa 1950. I noticed the definition was that of what I recognize to be the trace.

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