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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 ...
Conifold's user avatar
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14 votes
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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 ...
Conifold's user avatar
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12 votes
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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 ...
Mauro ALLEGRANZA's user avatar
12 votes
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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 ...
Torsten Schoeneberg's user avatar
11 votes
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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 ...
nwr's user avatar
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10 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 ...
kimchi lover's user avatar
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9 votes
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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'...
Conifold's user avatar
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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. ...
Francois Ziegler's user avatar
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 ...
J.G.'s user avatar
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7 votes
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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 ...
Jan Peter Schäfermeyer's user avatar
7 votes
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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, ...
Conifold's user avatar
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6 votes
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Were matrix theory and functional analysis well-known to physicists before the invention of matrix mechanics?

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 ...
Conifold's user avatar
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6 votes
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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 ...
linguisticturn's user avatar
5 votes
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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,...
Conifold's user avatar
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5 votes
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How did the exterior product get its symbol?

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) ...
Conifold's user avatar
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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 ...
paul garrett's user avatar
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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, ...
jpmarinier's user avatar
4 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 ...
Alexandre Eremenko's user avatar
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 ...
Mauro ALLEGRANZA's user avatar
4 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" ...
James H Davenport's user avatar
4 votes

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

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). ...
Gerald Edgar's user avatar
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4 votes
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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 ...
Jim Casey's user avatar
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 ...
user6530's user avatar
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4 votes
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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 ...
Conifold's user avatar
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4 votes
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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\...
Jean Marie Becker's user avatar
4 votes
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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....
Conifold's user avatar
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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, "...
paul garrett's user avatar
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4 votes
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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 ...
Cosmas Zachos's user avatar
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 ...
Alexandre Eremenko's user avatar
3 votes

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

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 ...
Margaret Friedland's user avatar

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