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As I searched for linear algebra, I found it odd that the linear map composition corresponds to the multiplication of matrices. Considering the intuition that the repetition of addition is multiplication, and the definition of a module, I thought it was better to call the scalar multiplication matrix multiplication. Of course, the scalar multiplication can be defined for any field, and people would have liked to call the operation between matrices "matrix multiplication". But I feel that matrix composition is a more logical term.

All I'd like to ask is about history. Was the multiplication of matrices "multiplication" from the beginning?

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    $\begingroup$ The answer to your second question can be no depending on where one places "the beginning". Cauchy (1815, p. 53) considers arrays he calls "systems" $$\begin{matrix}a_{1.1},&\dots&a_{1.n},\\\vdots&&\vdots\\a_{n.1},&\dots&a_{n.n},\end{matrix}$$ and calls our product $a\alpha^T$ the resultant of $a$ and $\alpha$ (pp. 78-79), proving (p. 82): le déterminant du système résultant est toujours égal au produit des déterminans des deux systèmes composans. $\endgroup$ – Consigliere ZARF Dec 14 '19 at 15:17
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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 (attending only to those of the same order) comport themselves as single quantities; they may be added, multiplied or compounded together, &c.: the law of the addition of matrices is precisely similar to that for the addition of ordinary algebraical quantities; as regards their multiplication (or composition), there is the peculiarity that matrices are not in general convertible; it is nevertheless possible to form the powers (positive or negative, integral or fractional) of a matrix, and thence to arrive at the notion of a rational and integral function, or generally of any algebraical function of a matrix".

Later, he first defines addition and multiplication by scalars, and then, symbolically introduces what is now called "row-column rule":

"...as the rule for the multiplication or composition of two matrices. It is to be observed, that the operation is not a commutative one; the component matrices may be distinguished as the first or further component matrix, and the second or nearer component matrix, and the rule of composition is as follows, viz. any line of the compound matrix is obtained by combining the corresponding line of the first or further component matrix successively with the several columns of the second or nearer compound matrix."

"Composition" and "multiplication" are used interchangeably throughout the paper. He goes on to define "inverse or reciprocal", and prove what is now called the Cayley-Hamilton theorem.

Cayley's use of "multiplication" should not be too surprising. Algebraic systems alternative to the usual arithmetic were actively developed at the time, and the words "addition" and "multiplication" came to be used abstractly. This is exactly Cayley's context in defining his algebra of matrices and discussing associative and other laws by analogy to the arithmetical ones. Hamilton did it with quaternions in 1843, although part of his motivation was composing rotations. Boole in his Laws of Thought (1854) did not hesitate to call logical conjunction "multiplication", explaining

"it is not affirmed that the process of multiplication in Algebra, of which the fundamental law is expressed by the equation $xy=yx$, possesses in itself any analogy with that process of logical combination which $xy$ has been made to represent above; but only that if the arithmetical and the logical process are expressed in the same manner, their symbolical expressions will be subject to the same formal law. The evidence of that subjection is in the two cases quite distinct".

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  • $\begingroup$ Calling logical conjunction “multiplication” is straightforward indeed – the booleans with $(\vee)$ form after all a commutative semigroup, just like the integers do with $(\cdot)$. However, matrix multiplication is merely a semigroupoid operation, that's quite a different category (pun not intended). $\endgroup$ – leftaroundabout Dec 15 '19 at 1:09
  • $\begingroup$ @leftaroundabout Cayley originally introduces it only for square matrices, and discusses algebraic laws in that context. It is extended to rectangular ones later in the paper. And conjunction is idempotent and applies to things with no connection to numbers, so in other ways more dissimilar than matrix multiplication. $\endgroup$ – Conifold Dec 15 '19 at 1:25

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