# Tag Info

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The same person who introduced it, Cayley. Sylvester first used the term "matrix" (womb in Latin) for an array of numbers in 1848, but did not do much with it. Cayley started developing matrix algebra in 1855 and summarized his theory in A Memoir on the Theory of Matrices (1858). In the opening paragraphs he writes: "It will be, seen that matrices (...

11

According to the following linked text, Weierstrass defined his equivalent form in a memoir presented to the Berlin Academy in 1868, two years prior to Jordan's Traite being published. From Bartel L van der Waerden's A History of Algebra : On the history of Jordan Normal Forms, see Thomas Hawkins "Weierstrass and the Theory of Matrices" Archive for ...

11

Are you familiar with Michael J. Crowe's book, A History of Vector Analysis? While I haven't read the book this article is well worth a read, and it seems to be a good summary. Of course, vector analysis is the precursor to linear algebra, so it won't directly address your question. Crowe does discuss briefly Grassmann's Ausdehnungslehre, one of the roots ...

11

The theory of Linear Algebra, along with the associated concept of linear mapping, was named as "linear" by its creator, Hermann Graßmann, which he developed in his 1844 linear algebra manifesto, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], and also later in Die Ausdehnungslehre: ...

10

The quoted 1946 English edition of Pontryagin's 1938 book is not the first appearance of kernel. It's already in the first English edition (1939). It's earlier in e.g. 1938 papers of J. H. C. Whitehead (p. 703) and Montgomery-Zippin (p. 364). The original German Kern is in P. Alexandroff and H. Hopf, Topologie I (1935, p. 557), and as Jan Peter comments ...

9

This is specifically about the history of linear algebra, history of Matrices and determinants.

9

Exactly; see Eigenvalues : The prefix eigen- is adopted from the German word eigen for "proper", "inherent"; "own", "individual", "special"; "specific", "peculiar", or "characteristic". It was David Hilbert who introduced the terms Eigenwert and Eigenfunktion; see: David Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1904)...

8

In a now-deleted comment, Consigliere ZARF listed a number of papers published in Zeitschrift für Physik in the late 1920's that used this notation. The earliest was Pascual Jordan's 1927 "Über eine neue Begründung der Quantenmechanik", using the notation on pp.816-817; with about 10 other papers published in the following few years, all in the ZfP, all ...

7

I think that by the end of the 19th and the beginning of the 20th century many mathematicians felt the need for a unified theory of "linear things" because they realized that they were doing the same things over and over again in different fields, e.g. Maxime Bocher writes in his Introduction to the study of Integral Equations that the notion of orthogonal ...

7

It seems incredibly unlikely that Gauss was (in a serious way) accused of witchcraft for his contributions to linear algebra. Instead of trying to look for anecdotal evidence, we can reason from general historical context. Firstly, note that it is easily found on Wikipedia that witch-hunts were mostly a thing of the 17th century and before, and instances ...

6

Cajori gives the early uses of logical symbols in volume 2 of History of Mathematical Notations. Neither Boole nor Schröder used $\wedge$ and $\vee$ in Boolean algebra, but rather $\cdot$ (or blank) and $+$, the idea was to make it as "algebra looking" as possible. Peirce in his 1865 lectures was apparently first to give Boolean algebra its modern form, by ...

6

According to an entry attributed to John Aldrich in Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, the word "kernel" in the algebraic sense was first used in the 1946 translation of Pontryagin's book Topological Groups by Emma Lehmer. So kernels of group homomorphisms came first. An unrelated use as in "integral kernel" occurs earlier,...

6

Cayley defines the inverse $M^{-1}$ of a matrix in the very same note Remarques sur la notation des fonctions algébriques (1855) where he first introduces matrix multiplication. He emphasizes the fact that a linear system $Mx = \xi$ is then solved by $x = M^{-1}\xi$, and that seems motivation enough for him. Here, $x,\xi\in \mathbb{R}^n$ and $M\in GL(n,\... 6 The "gradient descent" algorithm was invented before the gradient. It is described in equivalent form by Cauchy in a 3-page paper in Comptes Rendus, Méthode générale pour la résolution des systèmes d'équations simultanées (1847). Here is an English translation. A good secondary source is Cauchy and the Gradient Method by Lemaréchal, who comments: "Cauchy ... 6 Yes, it seems that there are linguistic reasons1 why positive definite works better than positively definite. 1BTW, for that reason, I think that it was a mistake to migrate this question from the English Language and Usage (EL&U) StackExchange to the History of Science and Mathematics (HSM) StackExchange. It seems that when we are picking adjectives ... 5 I am not sure what the editors of Wikipedia had in mind when arranging the names (if anything). Linear algebra textbook authors have them arranged every which way for both the formula and the identity, see e.g. Shafarevich-Remizov (p.69), Dym (p.103) and Lancaster (p.39). The formula was discovered independently but almost simultaneously in 1812 by both ... 5 From your sentence "my main motivation is still to try and get over my intense phobia of norms and inner products" I conclude that you need first of all a good book in Linear algebra itself, rather then history of linear algebra. In English, I recommend the textbook of P. Lax. There is a nice book of Dieudonne Algèbre linéaire et géométrie élémentaire (there ... 5 One can probably say that the relevant parts of algebra were "known to experts", rather than "well-known", and the relevant parts of functional analysis did not exist at the time, see Moore's Axiomatization of Linear Algebra: 1875-1940. Even finite dimensional matrices were not exactly standard teaching item yet, although Cayley gave the definition of ... 5 The expression "linear mapping" can have different meaning in different contexts. In many high school textbooks$x\mapsto ax+b$is called a "linear function". I suppose, because the graph is a straight line. According to the modern terminology, which comes from linear algebra only$x\mapsto ax$should be called linear, while$x\mapsto ax+b$must be called ... 5 Yes, orthogonal matrices with complex entries appeared at least as early as 1900, in E. Cartan's classification of simple Lie algebras (and Lie groups). In many ways, the complex numbers could be replaced by any algebraically closed field of characteristic$0$. Thinking of complex orthogonal groups as real Lie groups ("forgetting" the complex structure) ... 4 According to the Oxford English Dictionary, the first use of the word 'adjugate' is by CL Dodgson (Lewis Carroll) Proc. Roy Soc 15(1866) at p. 151. He says "In doing so I shall use the word "adjugate" in the following sense:-if there be a square block, and is a new block be formed, such that each of its terms is the determinant of the complemental minor of ... 4 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. ... 4 For a detailed discussion of Grassmann's work see : Hans-Joachim Petsche, Hermann Grassmann Biography (2009), and see page 186 for the discussion of the formula : After basically applying the conceptual constructions from the first section of Extension Theory in a more general way to elementary magnitudes in the second section, Graßmann introduces, ... 4 Linear algebra originally developed because of applications to differential equations and mechanics. Usually a system of differential equations can be approximated with a linear system, and a linear system requires linear algebra to solve it. I mean the period 18-19 centuries, and mathematicians like Lagrange, Cauchy, Jacobi. Later in 19th century it found ... 4 Keith: The earliest usage we have been able to find is in a 1936 paper by Francis J Murray and John von Neumann, entitled "Rings of Operators"(Annals of Mathematics, Vol. 37, pp. 116-229). Paul Halmos later used it in the earliest published version of his "Finite Dimensional Vector Spaces," published by Princeton University Press in 1942 (see Appendix II-... 4 They did. A natural way to treat such matrices is to introduce an indefinite inner product on$\mathbb{C}^n$, a non-degenerate bilinear form$(z,w):=z_1w_1+z_2w_2+\dots z_nw_n$, instead of the usual sesquilinear one. Then$A^T=A^{-1}$is equivalent to$(Az,Aw)=(z,w)$, i.e. complex orthogonal matrices are isometries of this space. The "orthogonal"/"unitary" ... 4 I will begin by 2 examples where the notion of dimension is essential : The increasing sequence of subspaces $$\{0\}\subset\ker(A-\lambda I)\subset\ker(A-\lambda I)^2\subset\ker(A-\lambda I)^3\subset\dots$$ (with ultimate stabilisation at a certain step) and the connection of their dimensions with the so important Jordan decomposition of a matrix$A$... 3 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 ... 3 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, ... 3 See Hawkins (1977, §5): 2$\times$2 case in (1868, pp. 111–113), n$\times\$ n in Traité des substitutions et des équations algébriques (1870, pp. 114–126). Also Cours d’analyse, vol. 3 (1887, pp. 173–175).

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