15
votes
Accepted
Who started calling the matrix multiplication "multiplication"?
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 ...
- 80k
14
votes
Accepted
What was the role of Schmidt in derivation of the Gram-Schmidt process?
A detailed history can be found in Gram-Schmidt orthogonalization: 100 years and more by Leon, Björck and Gander, see also their slides for a brief version. In short, Schmidt's 1907 presentation was ...
- 80k
12
votes
Accepted
Where does the name eigenvalue come from?
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 ...
- 15.2k
12
votes
Accepted
First use of "Spur" (trace) for linear maps / matrices
Comments by user njuffa (Thank You!) lead me to what I believe is what I was looking for:
R. Dedekind: Über die Discriminanten endlicher Körper. In: Abhandlungen der Königlichen Gesellschaft der ...
- 1,012
11
votes
Accepted
Who discovered the Jordan normal form?
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 ...
- 7,237
11
votes
Who introduced the "dagger"symbol as conjugate transpose in quantum mechanics?
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 ...
- 2,697
10
votes
Accepted
Who invented the gradient descent algorithm?
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'...
- 80k
9
votes
What came first? The kernel from vector spaces or from group theory?
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. ...
- 7,809
8
votes
Accepted
Motivation for the development of modern linear algebra
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 ...
- 1,404
8
votes
Why were the matrix multiplication rules chosen this way?
As with your previous question about why tessarines' commutativity doesn't win them more interest, commutativity is not mathematicians' goal; problem-solving is. We're interested in vector spaces, and ...
- 1,740
7
votes
Accepted
How and by whom was the concept of generalized eigenvectors developed?
This is somewhat difficult to track because much of the work on linear algebra in 19th century was coached in the language of anything but matrices and vectors, differential equations, substitutions, ...
- 80k
7
votes
Accepted
Origin of Multi-linear map
A pop pseudo-definition of multilinearity: A map is multi-linear if it is linear in each of its "components".
It's a very complicated story so this is -- despite its length -- a sketch at ...
- 3,139
6
votes
Accepted
Why do we call it a "positive definite matrix" rather than a "positively definite matrix"?
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 ...
- 475
5
votes
Accepted
Binet-Cauchy or Cauchy-Binet?
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,...
- 80k
5
votes
Have orthogonal complex matrices appeared in the literature?
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 ...
- 1,096
5
votes
Motivation for the development of modern linear algebra
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 ...
- 51.1k
5
votes
The terminologies "Adjoint" and "Adjugate"
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" ...
5
votes
Accepted
Why is the term "isotropic" used to describe a quadratic form and a vector?
This is an example of how a term migrates from the original context by broken telephone through various generalizations and transfers. It started with Poncelet introducing "imaginaries", i....
- 80k
5
votes
Earliest real-world uses of calculus and linear algebra
In the late 18th century, ships leaving Europe for other continents routinely brought with them books such as trigonometrical tables and astronomical almanacs.
These almanacs included ephemerides, ...
- 151
4
votes
Grassmann Formula
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 ...
- 15.2k
4
votes
Accepted
Have orthogonal complex matrices appeared in the literature?
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 ...
- 80k
4
votes
Why do we call it a "positive definite matrix" rather than a "positively definite matrix"?
As far as I know, the first appearance of the concept of positive/negative definiteness (and of indefiniteness) is in the article 271 of Gauss' Disquisitiones Arithmeticae about ternary forms. Of ...
- 4,400
4
votes
Accepted
First appearance of tensor product symbol $\otimes$
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 ...
- 56
4
votes
Accepted
What problem was solved by introducing the dimension of a vector space?
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\...
4
votes
When did linear algebra become the study of vector spaces?
Actually, I think it is not quite the case that what we now think of as "linear algebra" (either terminology or canon of standard results) existed in the late-ish 19th century.
Yes, "...
- 1,096
4
votes
Accepted
Who, between Cayley and Hamilton, first worked on the theorem that bears their name?
To avoid an opinion-based answer, one should focus on the first paper on the subject, (1858), by Arthur Cayley, "A Memoir on the Theory of Matrices", Philos Trans 148, preceding Hamilton's ...
- 1,949
4
votes
Earliest real-world uses of calculus and linear algebra
Applications of calculus actually predate its formal invention. For example, computation of volumes. One can argue whether Archimedes computations of volumes and areas have "real world ...
- 51.1k
3
votes
Accepted
Jordan's Paper on the Jordan Canonical Form
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–...
- 7,809
3
votes
Were matrix theory and functional analysis well-known to physicists before the invention of matrix mechanics?
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 ...
- 1,404
3
votes
Why were the matrix multiplication rules chosen this way?
Matrices were discovered in China in the 2nd century BCE as documented in the Nine Chapters of the Mathematical Art. Here, the method of Gaussian Elimination is used to solve matrix equations - and ...
- 3,760
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