# When exactly (and why) did matrices become a part of the undergraduate curriculum?

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 :-)

A similar question is posted on MO.

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-boggling" 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 matrices 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) matrices were taught earlier at the undergraduate level in history. That would be the United States in the 1950s.

• Can you give any reference(s) confirming that matrices were introduced to curriculum after WW II? Nov 1, 2014 at 23:48
• @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. Nov 1, 2014 at 23:53
• In what country was your father an engineering professor? Nov 1, 2014 at 23:58
• 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. Nov 2, 2014 at 0:36
• 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. Nov 2, 2014 at 16:29

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.

First a little history

The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.

This is discussed in Roger Hart, The Chinese Roots of Linear Algebra; however, in Euope

Systems of linear equations arose with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. However, the first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693.

In fact, Leibniz considered there to be a theory of 'extension' or 'logical characteristic' but wasn't able to come up with a viable such theory; in 1844 a prize competition was instituted on exactly this problem; this was won by Grassmann who had entered an essay 'Geometrische analsye...' after being persuaded by Mobius to enter; this included foundational new topics of what is today called linear algebra.

It was around this time (actually 1843), that Hamilton discovered the quaternions which prompted the discovery of other hyper-complex systems and then five years later, the English mathematician, James Joseph Sylvester introduced the term matrix (which is Latin for womb); it was another English mathematician, William Clifford who combined both Grassmanns theory and the theory of hyper-complex systems into what are now known as Clifford algebras.

In the transition from the early quantum mechanics to conventional quantum mechanics, Heisenberg and Jordan rediscovered matrix multiplication in 1925 (although Connes says that this would be better understood through groupoids).

It was Emmy Noether and her school who pioneered the study of abstract algebraic structures per se placing them in a systematic foundation; and in 1930, Van der Waerden published his Modern Algebra which 'forever changed' how algebra was taught in universities.

I would consider that it was all of these developments that pushed the undergraduate curriculum to the consideration of abstract algebra per se and of intrinsic structures and not just quantum mechanics.

(On a personal note, matrix mathematics wasn't just taught at universities, I distinctly remember being taught matrix mathematics at school).

• Mozibur: See also MO's answers. Gauss explicitly used and multiplied matrices, just under another name (and in curly brackets). Mar 25, 2020 at 20:58
• @ConsigliereZARF: So Gaussian elimination wasn't named spuriously... Mar 26, 2020 at 20:59