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Let me tell what I know about this. It is well-known that Heisenberg invented matrix multiplication himself, in his great paper that is considered part of the foundation of quantum mechanics. This was in 1925, and the story is very well documented. Then very shortly after that Born and Jordan recognized that this is matrix multiplication, BECAUSE one of them had a course on "hypercomplex numbers" as a student.

The clear conclusion that I make of this is that in the first decade (when they were all students) of 20th century matrix multiplication was not taught to students on a regular basis in the best European universities.

The first edition of Courant-Hilbert was published in 1924. (I am not sure what was the standard course of mathematics for physicists before that, but probably Thomson-Tait, which has no matrices).

On the other hand at the present time matrices are taught to ALL (science) undergraduates; this is even more standard than Calculus (I judge from my experience in Soviet Union and USA, but I suppose this is the case everywhere).

So my questions are:

  1. When did this dramatic transition in undergraduate curriculum happen?

and even more interestingly:

  1. Why did it happen?

On the second question I have a conjecture: it is exactly because of the invention of quantum mechanics. I have some supporting evidence and "philosophical arguments" in favor of this. But to investigate this matter, it is good first of all to find the answer on the first question.

I know that matrix multiplication was probably introduced by Cayley, but it is a very long way for a new mathematical object to the undergraduate curriculum, and most of our inventions never make it this way :-)

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I will start by answering why matrix algebra became important, and then discuss approximately when.

"Matrices" underpin what is often called operations research. That is, the theory of decision making. They are particularly useful in computer science, which features strings, arrays, etc., with machines substituting for human beings in (mechanical) decision making.

Operations research took a giant step forward during World War II, when the quantity of men, materials, weaponry etc. were "mind-boogling" for their time. As my father, a retired engineering professor would say, numerous "systems of equations" needed to be solved. (His first job out engineering school was to design an airfield.) During the war, the British government had some 1000 people in their "operational research" department, and likewise for the U.S. Some ten members of the U.S. group went to Harvard Business School together, then "parachuted" into Ford Motor Company as the "whiz kids."

So "matrices" was introduced into the undergraduate curriculum not long after World War II. The subject was given a boost by the newly-developed technique of "linear programming" (1947), followed by other decision-making tools such as input-output tables, which Wassily Leontief popularized in 1953. By the mid-1950s, "matrices" were taught at most of the better colleges, and by the late 1960s, they were finding their way into the high school curriculum.

It's true, as some commenters pointed out, that matrixes are now taught earlier in secondary school in countries outside the United States than "here." But that wasn't the question, which was about when (and where) matrixes were taught earlier at the undergraduate level in history. That would be the United States in the 1950s.

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    $\begingroup$ Can you give any reference(s) confirming that matrices were introduced to curriculum after WW II? $\endgroup$ – Alexandre Eremenko Nov 1 '14 at 23:48
  • $\begingroup$ @AlexandreEremenko:That's something I learned from "my father, a retired engineering professor." And the "correlation" with linear programming and input-output tables is strong. $\endgroup$ – Tom Au Nov 1 '14 at 23:53
  • $\begingroup$ In what country was your father an engineering professor? $\endgroup$ – Alexandre Eremenko Nov 1 '14 at 23:58
  • $\begingroup$ He's an American, but built the airfield in China (for the "Flying Tigers.") You and I are about the same age, and can remember studying matrices in high school in the late 1960s and early 1970s. $\endgroup$ – Tom Au Nov 2 '14 at 0:36
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    $\begingroup$ Both the question and answer are US specific. We were being taught matrix multiplication at high school in the UK (1980s). This included geometry but hints at operational research, linear programming, and (advanced level) eigenvalues would reinforce the above answer. $\endgroup$ – winwaed Nov 2 '14 at 16:29
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I would say that in Germany there was a gradual development towards the matrix notation of linear equation systems from the 1920s onwards. Courant certainly was a pioneer in this development as he tells in this interview.

This textbook from 1927 on Statik im Eisenbetonbau, i.e. statics of concrete structures, features the term "matrix" 65 times and was surely not inspired by quantum mechanics, but by the simplicity of matrix notation of the large linear equation systems that occur in structural mechanics.

From 1950 onwards matrices were taught in all technical and scientific disciplines at German universities as can be seen from this textbook by Zurmühl that went through three editions within 10 years.

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