# Questions tagged [linear-algebra]

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

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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 ...
227 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"?
162 views

### Why the matrix multiplication rules were 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 ...
222 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 ...
116 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"?
138 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 ...
270 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 ...
64 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 ...
172 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 ...
137 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 ...
344 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 ...
69 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 ...
64 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 ...
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 ...
1k 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 ...
316 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 $A,B$. (...
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 ...
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 ...
475 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?
905 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 ...
130 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.
76 views

### Jordan's Paper on the Jordan Canonical Form

In which paper, did Jordan introduce/prove the Jordan canonical form?
301 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/...
111 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 ...
528 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 ...
207 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 ...
1k views

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"...
212 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?
911 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 ...
593 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 ...
4k 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"?
540 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 ...
177 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 ...
277 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:...
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 ...