# Were matrix theory and functional analysis well-known to physicists before the invention of matrix mechanics?

Were matrix theory and analysis well-known to physicists circa 1920-1925? Did physicists make extended use of this theory in that period? The question is related to the discussion in How did Heisenberg come up with matrix mechanics? thread on Physics SE

One can probably say that the relevant parts of algebra were "known to experts", rather than "well-known", and the relevant parts of functional analysis did not exist at the time, see Moore's Axiomatization of Linear Algebra: 1875-1940.

Even finite dimensional matrices were not exactly standard teaching item yet, although Cayley gave the definition of matrix multiplication and developed some spectral theory back in 1850s, and Burali-Forti and Marcolongo published a book called Transformations Lineaires in 1912, which opens with:“We briefly set forth the foundations of the general theory of linear systems and linear operators. Generally, these matters are familiar in large part”. The ideas started percolating among physicists after the use of tensors in Einstein's general relativity, and Weyl's book on it Space, Time and Matter (1918) even introduces axiomatic vector spaces, inner product and congruence-preserving transformations in them. That Born, who in 1904 studied in Göttingen under Hilbert and Minkowski and returned there in 1921, was familiar with matrices and linear transformations is not therefore surprising. Neither rotations nor Lorentz transformations commute. But connecting the idea to infinite matrices was more analogy and physical intuition than applying an established mathematical theory. It is equally not surprising that young Heisenberg was not familiar with it, his 1925 paper does not even mention matrices, see Understanding Heisenberg’s ‘Magical’ Paper.

Hilbert introduced the "Hilbert space" in connection with integral equations beginning in 1904, but without treating them geometrically. Schmidt in a chapter "Geometry in a Function Space" (1908) wrote "For the geometric significance of the concepts and theorems developed in this chapter I am grateful to Kowalewski. It stands out even more clearly if $A(x)$ is defined, not as a function, but as a vector in a space of infinitely many dimensions". Riesz comes closest in his book (1913) on systems of linear equations with infinitely many variables, where he introduces the notion of orthogonal basis. In 1920-1922 Hahn, Banach and Wiener introduced normed linear spaces. However, this work did not study operators on infinite-dimensional spaces much, let alone presented them as infinite matrices. Such themes only emerge in von Neumann's works after 1927, and were motivated by quantum mechanics.

• The second part of Schmidt's paper deals with infinitely many equations in infinitely many unknowns and its main results are reproduced in this paper by Bocher and Brand: jstor.org/stable/1968085?seq=1#page_scan_tab_contents Feb 16, 2017 at 0:10

I heard a lecture by Heisenberg once, long ago. (A public lecture at MIT in the early 1970's.)

He remarked that he came up with a new, strange, sort of multiplication (that was not commutative). But then found from his colleagues that mathematicians had already been using it for 100 years. If Heisenberg wrote memoirs, presumably this is in there, too.

So this is support for the side saying matrix multiplication was not well-known to Heisenberg.

While their reception was rather slow, between Cayley's works from 1840s and 1850s and the much later development of vector spaces and functional analysis, matrices were considered by mathematicians of the late 19th and early 20th centuries in relations to complex numbers, quaternions, bilinear forms, systems of linear equations and determinants. Elements of matrix theory appeared in advanced textbooks and monographs. The book Z historie linearni algebry" by Jind\v rich Be\v cva\v r (Matfyzpress, Prague, 2007) mentions e.g. the following which appeared before 1925:

Ernesto Pascal: I determinanti. teoria ed applicazioni (1897).

Eugen Otto Erwin Netto: Vorlesungen \"uber Algebra (1896); Elementare Algebra. Akademische Vorlesungen f\"ur Studierende der ersten Semester (1904); Die Determinanten (1910).

Heinrich Weber: Lehrbuch der Algebra (second edition, 1898-99).

Alfred North Whitehead: Treatise on universal algebra (1898).

Leopold Kronecker: Vorlesungen \"uber die Theorie der Determinanten (1903).

Salvatore Pincherle: Lezioni di algebra complementare (1906-1909).

Maxime B\^ocher: Introduction to higher algebra (1907).

Cuthbert Edmund Cullis: Matrices and Determinoids (1913, 1918, 1925).

Leonard Eugen Dickson: Algebras and their Arithmetics (1923).

In Polish, there was an academic textbook by W\l adys\l aw Zaj\c aczkowskiThe Principles of Higher Algebra” (1884), which,among other things,presented the theory of determinants and of algebraic equations. There was also a wealth of material in the monograph of J\'ozef Puzyna on analytic functions (1898, 1900), including resultants and discriminants, binary forms, and the modular group.

It was not necessary to go to Goettingen to be exposed to matrix theory before 1925 (apparently, it was not sufficient either- Heisenberg studied there). Indeed, in his Nobel lecture in 1954, "The Statistical Interpretation of Quantum Mechanics ", Max Born stated explicitly:

This was in the summer of 1925. Heisenberg, plagued by hay fever took leave for a course of treatment by the sea and gave me his paper for publication if I thought I could do something with it. The significance of the idea was at once clear to me and I sent the manuscript to the Zeitschrift f\"ur Physik. I could not take my mind off Heisenberg’s multiplication rule, and after a week of intensive thought and trial I suddenly remembered an algebraic theory which I had learned from my teacher, Professor [Jakob] Rosanes, in Breslau. Such square arrays are well known to mathematicians and, in conjunction with a specific rule for multiplication, are called matrices. "

At that time, physics students were normally not taught linear algebra in Germany, except when their professors had an affinity to this subject. Max Born's teacher had been Jacob Rosanes, an algebraist, among whose students were Steinitz and Toeplitz. Heisenberg, on the other hand, had studied in Munich with Lindemann. Perron only came to Munich in 1922, when Heisenberg had almost finished his studies. Göttingen was an exception in Germany, and Heisenberg later stated that he had learned "optimism from Sommerfeld, physics from Bohr, and mathematics in Göttingen."

Here is an analysis of Heisenberg's breakthrough paper from July 1925: https://arxiv.org/pdf/quant-ph/0404009.pdf