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Questions tagged [linear-algebra]

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

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Earliest real-world uses of calculus and linear algebra

I want to illustrate in class that real-world applications of mathematics might take time to come to fruit. In this context, I want to find what the earliest real-world applications of Calculus and ...
Jaikrishnan's user avatar
2 votes
0 answers
73 views

Who discovered that the Lanczos method can only calculate extremal eigenvalues of large matrices?

The Lanczos tri-diagonalization process is widely or even routinely used today. It is said that it is useful for obtaining the extremal eigenvalues, but useless for medium eigenvalues. But who ...
poisson's user avatar
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3 votes
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What can I read to learn the history of multivariable calculus?

People have been doing calculus of several variables since well before the concepts of vectors, matrices, and linear algebra were formalized. Where can I learn about the development of multivariable ...
Dominic Stewart-Guido's user avatar
4 votes
0 answers
68 views

Sparse matrix ("matrice creuse") etymology in French

I am looking for the etymology of matrice creuse. According to Wikipedia, it seems James Joseph Sylvester used the term "matrix" in 1850, and Harry Markowitz used the term "sparse ...
Fnifni's user avatar
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6 votes
1 answer
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What was the role of Schmidt in derivation of the Gram-Schmidt process?

When reading the section related to Gram-Schmidt process in the book Linear Algebra and Its Applications by Gilbert Strang, I found a foot note that says: If Gram thought of it first, what was left ...
Tran Khanh's user avatar
2 votes
1 answer
84 views

Negative coefficients in the barycentric calculus

The barycentric calculus of Möbius involves formal sums of expressions of the form $mP$ where $m$ is a real number and $P$ is a point, where $mP$ is to be thought of as $m$ units of mass located at $P$...
James Propp's user avatar
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0 answers
57 views

Understanding Grassmann's Approach to Algebras over Vector Spaces?

According to Hermann Grassmann and the creation of linear algebra by Desmond Sander, Grassmann was able to identify all the important notions in linear algebra in his book "Ausdehnungslehre"....
Bumblebee's user avatar
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6 votes
1 answer
235 views

Earliest mention of permutation matrices, or equivalent? More generally, matrices for arbitrary functions between finite sets?

Permutation matrices I assume have a long history, and would be surprised if they were first considered only long after the work of Shur just after 1900, on the representation theory of $S_n$. ...
David Roberts's user avatar
3 votes
0 answers
90 views

Sylvester's Quote on Determinants

What does the following quote by Sylvester mean? "A general algebraical determinant in its developed form may be likened to a mixture of liquids seemingly homogeneous, but which, being of ...
stoic-santiago's user avatar
3 votes
0 answers
97 views

Where does the "operator to the right" notation originate?

If any of you have ever written code in DirectX, you're sure to have noticed that applying a linear operator $A$ to a vector $x$ is done as $xA$, instead of the (nowadays usual) $Ax$. I wanted to know ...
Francisco José Letterio's user avatar
2 votes
0 answers
147 views

Why did Clifford Algebra suddenly gain a resurgence under the name of Geometric Algebra in recent years?

To my understanding, this new Geometric algebra thing is actually nothing more than years old Clifford Algebra. Yet it is advertised by many of its proponents as a fundamentally new thing; why is that?...
tryst with freedom's user avatar
5 votes
1 answer
176 views

How and by whom was the concept of generalized eigenvectors developed?

In general, a linear operator on a complex vector space need not always have an eigenspace decomposition. But it will always have enough generalized eigenvectors to provide a decomposition of the ...
User1865345's user avatar
3 votes
0 answers
134 views

What motivated Grassman to study Grassman algebras in 1844?

These are $n$-tuples of reals, added componentwise and multiplied via the “exterior product.” They were introduced by Grassmann in 1844 as part of a brilliant attempt to construct a vector algebra in $...
tryst with freedom's user avatar
3 votes
0 answers
272 views

Origin of the usage of $\lambda$ to represent eigenvalues

I'm curious whether anyone knows how $\lambda$ came to be used to represent eigenvalues and or who (if anyone) was responsible for the convention. I've looked through a couple of books on the history ...
Scott H.'s user avatar
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15 votes
1 answer
408 views

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 ...
Torsten Schoeneberg's user avatar
5 votes
1 answer
151 views

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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6 votes
3 answers
825 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 “...
nwr's user avatar
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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 ...
pk123's user avatar
  • 51
2 votes
0 answers
97 views

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 ...
AChem's user avatar
  • 4,079
5 votes
1 answer
232 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, ...
Spencer Dembner's user avatar
4 votes
0 answers
152 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 ...
Wrzlprmft's user avatar
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2 votes
0 answers
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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 ...
Alexey's user avatar
  • 241
3 votes
2 answers
375 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 ...
Anixx's user avatar
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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 ...
Myvh's user avatar
  • 11
3 votes
0 answers
251 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 ...
ho boon suan's user avatar
0 votes
0 answers
89 views

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 ...
Mozibur Ullah's user avatar
7 votes
1 answer
766 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"?
Saikat's user avatar
  • 348
1 vote
0 answers
84 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 ...
GodTaxist's user avatar
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 ...
user2554's user avatar
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2 votes
3 answers
366 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 ...
Max's user avatar
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4 votes
0 answers
273 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 ...
roymend's user avatar
  • 123
10 votes
1 answer
4k views

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 ...
Manas Dogra's user avatar
4 votes
2 answers
176 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 ...
modnar's user avatar
  • 575
1 vote
1 answer
125 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"?
modnar's user avatar
  • 575
7 votes
2 answers
356 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 ...
modnar's user avatar
  • 575
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 ...
ististyle's user avatar
  • 303
9 votes
1 answer
1k 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?
M.M's user avatar
  • 193
2 votes
1 answer
95 views

Jordan's Paper on the Jordan Canonical Form

In which paper, did Jordan introduce/prove the Jordan canonical form?
user109871's user avatar
0 votes
1 answer
147 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.
aircraft's user avatar
  • 101
6 votes
1 answer
363 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/...
KCd's user avatar
  • 5,647
0 votes
1 answer
154 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 ...
Widawensen's user avatar
4 votes
2 answers
283 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 ...
Vladimir Lysikov's user avatar
3 votes
1 answer
250 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?
Stephan Kulla's user avatar
6 votes
2 answers
415 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 $A,B$. (...
Semiclassical's user avatar
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 ...
Christian's user avatar
  • 431
9 votes
1 answer
7k 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"?
Christian's user avatar
  • 431
4 votes
2 answers
736 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 ...
Elías Guisado Villalgordo's user avatar
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 ...
Alexandre Eremenko's user avatar
7 votes
5 answers
856 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 ...
Frobenius's user avatar
  • 173
4 votes
1 answer
276 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 ...
user2554's user avatar
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