I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today (as an introduction and overview of their place in mathematics). So I've been entertaining myself with reading the early works by Diananda, Motzkin, Gaddum and Hall & Newman and others. However, the most 'original' paper of them all, 'Copositive quadratic forms' by Motzkin from 1952, which is the very first paper discussing the concept, I can't find anywhere. I'm very interested in reading it, in order to find out why Motzkin considered them and how they fit in with his other interests. I also find that these early papers are quite inspiring for new ideas and since Motzkin was an especially talented mathematician I am wondering what his first approach was. Help would be appreciated!
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$\begingroup$ it appears that the NBS report series (where this work was published) has only one Motzkin paper digitized, not the one you want. $\endgroup$– Carlo BeenakkerCommented Feb 21, 2021 at 13:51
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1$\begingroup$ Motzkin has an abstract from 1965 which seems related: ams.org/journals/notices/196502/196502FullIssue.pdf (it is one paragraph, in p.44 of the pdf). It ends with "(See also, and amend, NBS Report 1818 {1952), 11-12.)" Maybe there was a mistake in the 1952 report? $\endgroup$– Geva YashfeCommented Feb 21, 2021 at 17:15
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$\begingroup$ It appears that whatever his reason was to consider copositivity, his work was ultimately not published elsewhere than in that report of the activities of his research group in 1952. The report he refers to does however note the characterization that he mentions, which means that he came up with the result at that time. I'm actually also interested in the actual paper corresponding to the abstract in your link, maybe more is to be found there! $\endgroup$– IAnemaetCommented Feb 21, 2021 at 17:47
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Hall and Newman (1963) cite this work as
Motzkin, T., Copositive quadratic forms. National Bureau of Standards Report 1818 (1952), pp. 11–12.
This cited part of the NBS report is available online at this link on pages 259-260. Please notice however that it's not a standalone work, but rather a review of other (possibly unpublished) manuscripts.
answered Feb 21, 2021 at 14:06
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$\begingroup$ when I searched for the report in the archive I cited in my comment I could not find the Motzkin paper; it should be the report number 1818 at this link -- but that returns a "not found", hence my conclusion that the 1818 report has not been digitized; the other Motzkin report (number 1902) does return a scan , no idea why the 1818 report is missing. $\endgroup$ Commented Feb 21, 2021 at 15:54
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2$\begingroup$ @CarloBeenakker: At the website I linked, the report number 1818 starts at page 243. $\endgroup$ Commented Feb 21, 2021 at 17:17
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1$\begingroup$ very glad to read this! (Also note the work of Cornelis Lanczos a bit earlier in this report!) So it seems Motzkin wanted to derive a characterization for the existence of a solution of linear inequalities. He refers at one point to '(1)', but I can't find the equation he refers to anywhere. I assume that he meant the system Ax >= 0. Just fascinating! $\endgroup$– IAnemaetCommented Feb 21, 2021 at 17:50
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$\begingroup$ thank you for the explanation! $\endgroup$ Commented Feb 21, 2021 at 18:24
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1$\begingroup$ Just a little more investigation on my part: the (1) Motzkin refers to is the paper "Basic solutions of the transportation problem" which apparently never got published. The other papers did however get published (Blumenthal and Gaddum), and looking through them they seem to borrow Motzkin's result here and there! (Or make some very similar statements about minors and whatnot). $\endgroup$– IAnemaetCommented Feb 21, 2021 at 18:58