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History of “Gaussian” elimination
Joseph Grcar.YouTube,01:24:37,published onFeb 6, 2015,ICMEStudio,https://www.youtube.com/watch?v=KxmmYve4AX0.Type:Video
Date Reviewed: Jul 22 2016

The first interesting fact made clear in this talk is that Gaussian elimination, as we know it today, has its origins so far back in time that certainly in the beginning it was neither “Gaussian” nor “elimination.” Some of the basic procedures for dealing with simultaneous equations did involve some elimination of variables, but it was applied to areas, as was the case in the Ancient Middle East, rather than linear problems. It couldn’t be “Gaussian” because it appeared much before the time of Gauss. In fact, the first person to use it for both linear and nonlinear problems was Sir Isaac Newton, but he called it “extermination” of variables. Moreover, his notes were published in spite of his wishes.

This video is a review of the history of Gaussian elimination, with a focus on how it came to acquire such a name, and covering the main ideas of linear algebra involved, such as the concepts of systematic variable elimination, symbolic notation, equivalence with matrix operations, algorithmic procedures, and technology usage.

It can be said right away that Gauss became attached to the name of the algorithm because of his notation and algorithmic procedure that made it quite convenient when computers came into existence.

The word “elimination” was not used systematically until the mathematician Sylvester François Lacroix in the 19th century published as such, in a time when France was one the most dynamic places for mathematical development.

Gauss actually developed systematic elimination and bracket notation (akin to our modern indexed entries of matrices) for his least squares method, for the precise calculation of areas, when those areas were composed of triangles that often did not match precisely. It was through the development of the solution of an overdetermined linear system that Gauss could estimate areas.

As concerns technology, Grcar describes the evolution from the usage of logarithm tables, through pre-calculated multiplication tables, until the earlier mechanical machines--these last influenced the usage of the algorithm developed by Gauss, hence his name on it.

The audio and video of the speech are in general very good; one has no problems distinguishing the words and formulas projected on the screen, and discerning the audio is not a problem either, not only from a technical recording point of view but also because of the particularly good enunciation of the speaker.

For many of us who were taught about Gaussian elimination, knowing about its evolution in the way presented in this video is important as an inspiration on how we should ourselves present our work for posterity: the algorithm should not be too attached to current technology, but be efficient in a general way, and it should be shown being applied to an important problem. In the times of Gauss, it was map making; in the earlier 20th century, it was applied to electrical problems; nowadays, it is applied to digital.

This video is not just for specialists. It is so clear that it can (and should) be shown to early undergraduates, as they will probably gain the most from watching it. The conclusions that can be drawn from the video encourage viewers to be clear in the way they expose their novel work.

Thus, I recommend this video to both students and mathematicians in general, and encourage those interested to read its originating paper [1].

Reviewer:  Arturo Ortiz-Tapia Review #: CR144622 (1611-0833)
1) Grcar, J. F. How ordinary elimination became Gaussian elimination. Historia Mathematica 38, (2011), 163–218.
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