矩陣何時準確(以及為什麼)成為本科課程的一部分?


41

讓我告訴我有關這件事的知識。眾所周知,海森堡在他的偉大論文中發明了矩陣乘法,該論文被認為是量子力學基礎的一部分。那是在1925年,這個故事有很多記載。之後不久,博恩和喬丹意識到這是矩陣乘法,因為其中之一是作為一名學生開設了"超複數"課程。

我對此做出的明確結論是,在第一個十年中歐洲最好的大學不會定期向學生教授20世紀矩陣乘法的所有學生。

Courant-Hilbert的第一版於1924年出版。(我不確定在那之前物理學家的數學標準課程是什麼,但是可能沒有矩陣的Thomson-Tait是)。

另一方面,目前正在向ALL(理科)本科生教授矩陣;這甚至比微積分更標準(我根據我在蘇聯和美國的經驗判斷,但是我想到處都是這種情況。

所以我的問題是:

  1. 本科課程的這種戲劇性轉變是何時發生的?

更有趣的是:

  1. 為什麼會發生?

在第二個問題上,我有一個推測:正是由於量子力學。我對此有一些支持證據和"哲學論證"。但是要對此事進行調查,首先找到第一個問題的答案是一件好事。

我知道矩陣乘法可能是Cayley引入的,但是對於本科課程中一個新的數學對象來說這是很長的路要走,而且我們的大多數發明都從來沒有這樣:-)

類似的問題發佈在MO上。

27

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.


6

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.


0

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).