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Could anyone give any information about the invention of the concept of the trace of a Matrix, as this concept is so important and useful in linear algebra. I searched on the internet, but found nothing on its origin.

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  • $\begingroup$ Maybe one for hsm.stackexchange.com/questions ? $\endgroup$ Commented May 21, 2020 at 5:51
  • $\begingroup$ @AnginaSeng Thanks for your advice, I tried keyword such as "trace matrix" in HSM, no results so far. $\endgroup$
    – C. Davide
    Commented May 21, 2020 at 6:00
  • $\begingroup$ I think I would start with Muir's History of Determinants and the references therein. $\endgroup$ Commented May 21, 2020 at 6:25
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    $\begingroup$ I flagged this question so that hopefully a moderator can move it to History of Science and Mathematics SE. That’s not a bad thing! I’m not voting to close it or voting it down. I just think you’ll get a better answer there as this kind of question is their specialty. $\endgroup$ Commented May 21, 2020 at 6:59
  • $\begingroup$ @Moo Great thanks for these two links, basically what i am looking for, although the very first who used the German word "die Spur" is still unclear! $\endgroup$
    – C. Davide
    Commented May 21, 2020 at 13:13

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To upgrade Moo's answer with what, for all we know, is the German-language prehistory of "trace":

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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From Where did the German term "Spur" of a matrix come from? we learn of "die Spur". You can also review Motivation for the term "trace" in linear algebra.

Using that tidbit and this website, we find:

Trace (of a matrix) is a translation of the German die Spur (related to the English word "spoor.")

In his 1922 translation of H. Weyl’s Raum, Zeit, Materie (Space-Time-Matter) H. L. Brose writes: "the trace (spur) of a matrix."

Some writers in English preferred the term "spur," e.g. A. C. Aitken Determinants and Matrices (9th edition 1956) writes "the spur or trace of A. We shall denote it by sp A."

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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), phrasing it as if his audience might be acquainted with the concept and terminology. Since his main audience was Frobenius, looked further into his works, but from what I can tell, the word "Spur" in conspicuously absent from all his papers on group characters. In e-rara.ch/zut/doi/10.3931/e-rara-18879, he just calls it "the sum of diagonal elements". $\endgroup$ Commented Jun 27, 2022 at 5:33
  • $\begingroup$ @TorstenSchoeneberg: Very interesting! It also shows and reminds us why capturing history accurately is so difficult. $\endgroup$
    – Moo
    Commented Jun 27, 2022 at 10:57
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    $\begingroup$ I've asked an updated version of this at hsm: hsm.stackexchange.com/q/14563/6514 $\endgroup$ Commented Jun 27, 2022 at 22:45
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Cayley (in particular) , and other British Mathematicians, wrote early papers on matrices, he even used [a b, c d] as notation for a typical 2x2 matrix, which we still use today. So probably trace is in these early papers. Others involved were Sylvester, Kirkman.

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