Questions tagged [linear-algebra]

For questions about linear algebra, a mathematical field studying vector spaces and matrices.

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First use of "Spur" (trace) for linear maps / matrices

Every student of linear algebra learns about the trace of a linear map. Its easiest (albeit not most conceptual) definition is: write the map as matrix, then the trace is the sum of the diagonal ...
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58 views

Were 3-dimensional split-complex numbers ever described in literature?

Basically, if you add two complex dimensions to reals, say $i$ and $j$, you automatically get a fourth dimension $ij$ because this number cannot be expressed using only the three dimensions. The ...
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4 votes
1 answer
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Who, between Cayley and Hamilton, first worked on the theorem that bears their name?

I know that Frobenius is the one who proved the Cayley-Hamilton theorem in all its generality. However, between Cayley and Hamilton, who did first work on the subject? In English: Cayley–Hamilton ...
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4 votes
3 answers
230 views

When did linear algebra become the study of vector spaces?

All of the concepts and terminology central to linear algebra were established in the late 19th century. Following recent comments by user KCd, that in the early 20th century determinants were the “...
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4 votes
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Who proved Rank Nullity Theorem?

I have been learning about the Rank Nullity theorem and was trying to understand Who came up with the rank nullity theorem? While i did look up on the internet i came up with almost no answers. Some ...
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2 votes
0 answers
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History of circulant matrices for convolution

Discrete linear convolution $$ y[k]=h[n] * x[n]=\sum_{i=-\infty}^{\infty} x[i] h[k-i] $$ can be done with circulant matrices with appropriate zero padding. Is anyone aware of the name of the ...
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  • 2,989
3 votes
1 answer
109 views

Why is the term "isotropic" used to describe a quadratic form and a vector?

A quadratic form $q$ on a vector space $V$ is isotropic if $q(v) = 0$ admits a nonzero solution. What was this terminology originally intended to evoke? There is some prior discussion on Math SE, ...
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4 votes
0 answers
135 views

Who bet against the usefulness of matrix inversion – or is it a myth?

In my linear-algebra and numerics courses, I frequently heard an anecdote about some professor betting – literally, with money – that there would never be any application where computing the actual ...
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2 votes
0 answers
71 views

Origin of Lang's proof of the Cayley-Hamilton theorem

Is the proof of the Cayley-Hamilton theorem given by Serge Lang in Algebra (page 561) an original one, or has it been borrowed from some earlier sources? Who came up with it first? (Lang's proof is ...
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  • 191
3 votes
2 answers
306 views

Why were the matrix multiplication rules chosen this way?

The currently standard matrix multiplication is isomorphic to split-quaternions (for 2x2 matrices, and similar for higher ranks, maybe this is called Clifford algebras). As such, the sets of square ...
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  • 598
1 vote
0 answers
77 views

Why are linear forms called "forms"?

My question is about linear forms, quadratic forms, n-linear forms, differential forms and so on. The first term of these names seem clear to me, but I cannot make a link between these mathematical ...
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  • 11
3 votes
0 answers
158 views

The origins of $\det(I+AB)=\det(I+BA)$

I am looking for the earliest published source that gives and perhaps proves the identity $\det(I+AB)=\det(I+BA)$ where $A$ and $B$ are just rectangular matrices of finite dimensions (as opposed to ...
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0 answers
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What is the earliest article in which Leibniz used 'matrices'?

The Chinese were using matrices ( fengcheng in the Nine Chapters of the Mathematical Art), long before they were used in Europe which suggests that possibly they were introduced by way of them. For ...
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7 votes
1 answer
402 views

Origin of Tensor Product

When and why did Mathematicians saw a need to define Tensor Products? I want to know the historical development of the idea "Tensor Product"?
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1 vote
0 answers
72 views

Research about Stafford Beer's claim about a method for solving simultaneous equations unknowingly via a game by kids?

I found this claim in the book "How many grapes went into the wine", in the Artorga section: In 1956 I devised a game for solving simultaneous linear equations in two variables. The theory ...
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2 votes
0 answers
112 views

Interpretation of a short note of Gauss on the resolution of a special system of inhomogeneous linear equations by roots of unity

My question refers to a 2-pages fragment of Gauss, entitled: "Note on the resolution of a special system of linear equations", which is found on pages 30-31 of volume 8 of his works. In this ...
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2 votes
3 answers
299 views

What problem was solved by introducing the dimension of a vector space?

In linear algebra, we care a lot about dimensions. I get why it’s useful but not why it’s such a big deal. So I wondered what problem was solved historically by introducing a rigorous definition of ...
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4 votes
0 answers
191 views

Why did Jordan introduce his canonical form?

Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear ...
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  • 81
9 votes
1 answer
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Who introduced the "dagger"symbol as conjugate transpose in quantum mechanics?

The $\dagger$ symbol is often used in quantum mechanics,and also often in general mathematics to represent the conjugate transpose operation.For Hermitian matrices we can write $$A^\dagger=A$$Who ...
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4 votes
2 answers
158 views

Have orthogonal complex matrices appeared in the literature?

According to https://en.wikipedia.org/wiki/Orthogonal_matrix, https://en.wikipedia.org/wiki/Unitary_matrix, and Friedberg et al.'s Linear Algebra (4th edition), a matrix $A\in F^{n\times n}$ is ...
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  • 545
1 vote
1 answer
121 views

Why positive definite matrix rather than positively definite matrix? [duplicate]

"Positive definite matrix" is a standard term in mathematics, espeically linear algebra. Are there grammatical, linguistic, or historical reasons why it was not called "positively definite matrix"?
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  • 545
7 votes
2 answers
267 views

Why do we call it a "positive definite matrix" rather than a "positively definite matrix"?

The term positive definite matrix is a standard one used in mathematics, especially in linear algebra. Are there grammatical, linguistic, or historical reasons why it was not called a positively ...
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  • 545
13 votes
1 answer
1k views

Who started calling the matrix multiplication "multiplication"?

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 ...
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  • 241
7 votes
1 answer
634 views

Who invented the gradient descent algorithm?

In connection to the question "Who invented the gradient?", I want to know who invented the gradient descent algorithm?
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2 votes
1 answer
80 views

Jordan's Paper on the Jordan Canonical Form

In which paper, did Jordan introduce/prove the Jordan canonical form?
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0 votes
1 answer
143 views

How to understand `After quantum mechanics, nature itself suddenly became linear`?

How to understand Freeman Dyson's Saying: After quantum mechanics, nature itself suddenly became linear.
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  • 101
6 votes
1 answer
336 views

First appearance of tensor product symbol $\otimes$

I was asked recently if the tensor product symbol $\otimes$ had been used before Bourbaki's publication on multilinear algebra in 1948 (a draft of this document can be seen at http://sites.mathdoc.fr/...
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  • 4,064
0 votes
1 answer
121 views

Invention of matrix symbols

Who first did use the symbol $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ for a matrix and similarly $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$? Why do these two ...
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4 votes
2 answers
241 views

Where did block matrix multiplication appear?

I am curious about who first noticed that block matrices can be multiplied blockwise. There is a section about matrices partitioned into submatrices that describes block matrix multiplication in "An ...
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3 votes
1 answer
225 views

Why is the term "kernel" used in algebra? [duplicate]

What was the motivation to use the word "kernel" in algebra to denote the set of all arguments which are mapped to the idendity element (by a homomorphism)? Who introduced it?
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5 votes
2 answers
341 views

Binet-Cauchy or Cauchy-Binet?

The Cauchy-Binet formula in linear algebra gives the determinant of a square matrix $AB$ for rectangular matrices $A,B$ of appropriate shape in terms of the determinants of the submatrices of $A,B$. (...
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4 votes
3 answers
1k views

Motivation for the development of modern linear algebra

In many text books on linear algebra one of the most important applications is geometry in Euclidean spaces and therefore there seems to be the impression that linear algebra was developed in order to ...
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9 votes
1 answer
5k views

Where does the name eigenvalue come from?

Who introduced the concept of eigenvalues and eigenvectors and where does the name come from? Is there a connection with the German word "eigen"?
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  • 421
4 votes
2 answers
631 views

Grassmann Formula

I'm in my first year of Mathematics at the University. Recently, we've learnt about Grassmann Formula and when I was making a little research on the internet, I couldn't find a single reference ...
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4 votes
2 answers
1k views

Who discovered the Jordan normal form?

In the paper by A. J. Coleman, "The greatest mathematical paper of all time" (Math Intelligencer, 11, no. 3 (1989), 29-39), on page 30 there is a passing remark that the "Jordan form is due to ...
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7 votes
5 answers
667 views

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 ...
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  • 175
4 votes
1 answer
265 views

Did Leibniz sketch a design for a machine capable of solving a system of linear equations?

My question is based on the information on pages 108-109 of the book The Tangled Origins of the Leibnizian Calculus. I know that Leibniz invented the stepped drum and used it to build the stepped ...
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0 votes
0 answers
193 views

What questions led Cayley to the definition of matrix multiplication?

quote: every book I've seen on matrix algebra or linear algebra seem[s] to just define the matrix operations without providing any historical background Talk:Matrix multiplication - Wikipedia, the ...
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6 votes
2 answers
384 views

How did the exterior product get its symbol?

As per the title: where did the notation $a\wedge b$ for the exterior product of $a$ and $b$ originate, and/or who popularised it? I'm especially interested in motivation for the choice of this symbol ...
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1 vote
0 answers
298 views

Need information about the history of the Hotelling and Bodewig method

I need information about Hotelling and Bodewig, who they were and why they developed this method. anything will help, references to articles, links, or any other information. Link to the method: http:...
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4 votes
2 answers
1k views

The terminologies "Adjoint" and "Adjugate"

The adjoint of a matrix is also called the adjugate. However, the word "adjoint" is also used in Linear Algebra for linear transformations, and the "adjoint of a matrix" and "adjoint of a linear map"...
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  • 191
8 votes
3 answers
801 views

Why do we call a linear mapping "linear mapping"?

According to P. M. Cohn's Classic Algebra, for historical reasons we call a linear mapping "linear mapping". What are the historical reasons that led to the adoption of the term "linear ...
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  • 189
11 votes
2 answers
579 views

What came first? The kernel from vector spaces or from group theory?

In studying vector spaces we learn about linear transformations from one vector space to another and in particular the kernel of such a transformation. When learning about group theory we also learn ...
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7 votes
1 answer
2k views

History of the inverse matrix

I know the definition and the procedure to calculate the inverse of a matrix, but I want to know the history of starting the idea of an inverse matrix. mathematicians must have faced a real life ...
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8 votes
1 answer
395 views

Gauss accused of witchcraft: apocryphal?

I recall reading years ago in a linear algebra book that Gauss was accused of witchcraft for his (re)discovery of what we now call the row reduction algorithm for solving linear equations. Has anyone ...
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  • 1,021
19 votes
4 answers
2k views

Books on the history of linear algebra

I'm quite desperate to understand the historical motivation and origin of all of the "geometrical" concepts of linear algebra, namely: The concept of thinking of elements of $\mathbb R^n$ or some ...
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  • 3,019