# First use of "Spur" (trace) for linear maps / matrices

Every student of linear algebra learns about the trace of a linear map. Its easiest (albeit not most conceptual) definition is: write the map as matrix, then the trace is the sum of the diagonal entries.

The question about origin and motivation of this terminology has been asked several times: https://math.stackexchange.com/q/1291981/96384, https://math.stackexchange.com/q/2593048/96384, https://math.stackexchange.com/q/3684703/96384.

However, none of them have a fully satisfying answer. What they seem to agree on is that the English "trace" is a translation of the German terminology Spur (which is cognate to English spoor, but indeed translates to trace or track): "Earliest Known Uses of Some of the Words of Mathematics" quotes H. L. Brose's 1922 translation of a book of Weyl's, which might be the first use of the English term "trace". But then the question remains,

In what (presumably German-language) source was the word "Spur" used first for this concept?

Attempt: At the bottom of page 6 of his (1901) doctoral thesis Ueber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, Issai Schur writes somewhat casually [emphasis added]:

dass zwei invariante Formen $$T(A)$$ und $$T_1(A)$$ dann und nur dann äquivalent sind, wenn ihre Spuren, d.h. die Summen der Glieder ihrer Hauptdiagonalen einander gleich sind.

Since Schur's doctoral advisor was Frobenius, I thought it natural to look for earlier occurrences in Frobenius' works. However, I learned that in Frobenius's three Sitzungsberichte papers from 1896 where he founded character theory, he actually does it via a different route, and I found no trace of a "Spur" in those. (For some overview on the history of representation and character theory, cf. T. Hawkins and K. Conrad.) Then, in Frobenius' 1897 paper Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, where he finally uses matrices, he does note that his characters turn out to be sums of diagonal elements (equation 5 in §4), and he emphasizes the importance of this especially in §6 (pages 15 and 16), but as far as I can tell, nowhere does he use the word "Spur"; instead, he writes out "die Summe der Diagonalelemente" twice.

So now I am almost tempted to believe that Schur might have casually introduced that terminology there? Of course I might have overlooked something in Frobenius' work of that time. Another guess would be Dedekind, who according to the mentioned historical sources played an important "inspirational" role in that time. I have not looked through any of his works yet. Surely from today's view, he would have had reason to be interested in "Spuren" already in his number theoretic works.

PS: In one of the linked questions above, a theory is put forward and credited to P.J. Cohen that supposedly, German "Spur" was a mistranslation of an English word "spur" supposedly used by Cayley (which would have been earlier than my guesses). This theory sounds rather unlikely to me, and according to a comment by Brian M. Scott to that question, "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."

• But wait a few days (say a week) before posting elsewhere. There may be answers here. Jun 28, 2022 at 1:21
• R. Dedekind, "Über eine Erweiterung des Symbols $(\mathfrak{a} \mathfrak{b})$ in der Theorie der Moduln, Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1895, pp. 183-208 (scan online): "[..], daß man den endlichen Körper $\Omega$ nicht nur auf den Körper der rationalen Zahlen, sondern auch auf jedem in $\Omega$ als Divisor enthaltenen Körper bezieht, wobei neben den gewöhnlichen Normen, Discriminanten, Spuren, auch [...] Jun 28, 2022 at 7:30
• Heinrich Weber, Lehrbuch der Algebra, Zweite Auflage, Braunschweig: Vieweg 1898, p. 502 (online): "Es ist die Summe $S(\Theta) = \Theta + \Theta_{1} + \Theta_{2} + \ldots + \Theta_{n-1}$, die wir die Spur von $\Theta$ nennen, und das Product $N(\Theta) = \Theta \Theta_{1} \Theta_{2} \ldots \Theta_{n-1},$ das die Norm von $\Theta$ heisst." Jun 28, 2022 at 7:48
• The same sentences appears in the first edition of Weber's Lehrbuch (1895), p. 461. I have yet to find any mention of "Spur" in a publication by Frobenius, but am still searching. Jun 28, 2022 at 9:38
• Only paper authored by Frobenius that uses "Spur" that I could find is a joint publication with Schur, and it post-dates the paper cited in the question: G. Frobenius and I. Schur, "Über die reellen Darstellungen der endlichen Gruppen," Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1906, pp. 186-208 Jun 28, 2022 at 10:36

Comments by user njuffa (Thank You!) lead me to what I believe is what I was looking for:

R. Dedekind: Über die Discriminanten endlicher Körper. In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen (1882).

On page 5 he writes (and the emphasis is in the original):

Unter der Spur der Zahl $$\theta$$ verstehen wir die Summe aller mit ihr conjugirten Zahlen; wir bezeichnen diese offenbar rationale Zahl mit $$S(\theta)$$; dann ist $$(11) \qquad \qquad S(\theta)= \theta^{(1)} + \theta^{(2)} + \dots + \theta^{(n)} = -a_1\\ \qquad= e_{1,1} + e_{2,2} + \dots + e_{n,n}$$

That is, he is looking at the field trace of an algebraic $$\theta$$ over $$\mathbb Q$$, and in one swoosh gives the three classical definitions: as sum of all its Galois conjugates, as the negative of the next-to-leading term of its minimal polynomial, and as sum of the diagonal entries of a matrix representing it. He goes on to note that the trace is linear and $$S(1)=n$$, the degree of the field extension.

I believe this is Dedekind's invention and the first published use of the term, given the following extra evidence:

Dedekind had published "Supplement XI" to Dirichlet's Vorlesungen über Zahlentheorie in its third edition, 1879. This Supplement XI (founds modern algebraic number theory and) in this 1879 version, contains an introduction to norms and discriminants of number fields (§ 164), but "Spuren" are still absent. E.g. when on page 497 (§166) he goes through the example of a quadratic field $$\mathbb Q(\omega)$$ with $$\omega'$$ the conjugate of $$\omega$$, he calls the product $$\omega \cdot \omega'$$ by name and notation $$N(\omega)$$, but writes down the sum $$\omega +\omega'$$ next to it, without terminology or notation.

So I assume he amended the concept and terminology of "Spur" in the above 1882 article, in which it is used for further inquiries into the discriminant.

In fact, for the fourth edition of Dirichlet's Vorlesungen (1894), Dedekind had revised and enlarged "Supplement XI", and now the new §167 contains a treatment of norms, discriminants, and "Spuren", which incorporates a streamlined version of the 1882 article, and actually could be copied into a textbook on algebraic number theory today. (Already in the introduction to his 1882 article, Dedekind says that although he works with the ground field $$\mathbb Q$$ for now, completely analogous results would hold over other fields, i.e. proposes the relative trace of a field extension $$L\vert K$$; and he introduces it in this generality in the 1894 version.)

Also, njuffa found:

• An article of Hensel's from 1897 (Ueber die Elementartheiler zweier Gattungen, von denen die eine unter der anderen enthalten ist, Journ. f. d. reine u. ang. Math. Vol. 117, Jan. 1897, pp. 346-356, where he also uses the concept for general field extensions, and claims the terminology to be due to Dedekind ("Jede solche Grösse $$H_i$$ werde in Erweiterung einer Dedekindschen Bezeichnung die Spur von $$Y_i$$ genannt", p. 347).

• Weber's textbook Lehrbuch der Algebra (first edition 1895), where the Spur (together with the norm) is introduced on page 461 (§144), although only via the definition with conjugates, which thus must have been the main point of view at that time; so much so that, I assume, when Frobenius in 1897 realized that his characters are also sums of diagonal elements, he ignored that this was an alternative definition of a somewhat established concept in number theory.

But, as pointed out by KCd in a comment, Frobenius must have seen this shortly after, as 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, [...]

("Vorgang" which nowadays translates as "procedure" is, to my German sense, a bit of an archaic word for "precedent" here, i.e. in modern language he says "If one, following Dedekind, calls the sum ...")

And certainly with and after this, and Schur's 1901 dissertation cited in the OP, the door was open for the trace of general linear maps / matrices, not just ones coming from elements in field extensions.

• Very good findings. I am still wondering about the reason for calling the sum of diagonal elements as "Spur" or "trace"? Jun 29, 2022 at 19:06
• All speculation, but for Dedekind the sum of diagonal elements was probably more a computational afterthought. He was defining, after already knowing norms and discriminants, a new rational invariant of an algebraic number, which sometimes vanishes (e.g. for $\sqrt{-1}$ or $\sqrt[3]{5}$) and sometimes does not (e.g. for $4+3\sqrt{2}$). To call such an invariant the element's "trace" (which it leaves in the ground field, for us to see and get a first idea of the element) seems not totally unmotivated. Jun 29, 2022 at 20:00
• The index of Hawkins' The Mathematics of Frobenius in Context has two entries for trace: "term introduced by Dedekind" and "term popularized by Frobenius", both on page 500, where it says trace "was not a common part of a mathematician's vocabulary at the end of the 19th century" and that Frobenius began using it in 1899 (so your guess 1897-1901 is right) in Ueber die Composition der Charaktere einer Gruppe, Sitzungsbreichte der Königlich Preuss. Akad. der Wiss. zu Berlin (1899), pp. 330–339. See biodiversitylibrary.org/item/93034#page/378/mode/1up, end of 1st page.
– KCd
Jul 1, 2022 at 20:18