Questions tagged [linear-algebra]
For questions about linear algebra, a mathematical field studying vector spaces and matrices.
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A modern reinterpretation of Vandermonde's mathematical achievements
I already posted a question on the Mathematics Stack Exchange about a related topic, but after posting it, I realized that it is more appropriate for this site, so I am posting it again in this site.
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Why did Vandermonde become interested in algebraic problems?
I recently became interested in a person named Vandermonde when I posted a question on this site about the history of determinants and read the answers and comments. He was a person who loved music ...
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Origins of the use of matrices for the calculus of binary relations
Nowadays it's common to represent a binary relation $r$ (a subset of the Cartesian product of two sets $A$ and $B$) as a 0,1-matrix whose rows correspond to elements of $A$, whose columns correspond ...
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How was Laplace able to propose the Laplace expansion?
I read Sylvester's 1850 paper, which is on file here. If you look at page 147 of this file, you will see the following sentence:
Imagine any determinant set out under the form of a square array of
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History of Invertible linear transformations
Who studied invertible linear transformations for the first time? I would guess these linear transformations weren't studied for the sake of studying invertible linear transformations, they probably ...
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Origin of decomposition theorem
There are many different types of decomposition theorem in linear algebra. For example, there are primary decomposition theorem, cyclic decomposition theorem, etc. But I became curious about the ...
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Origin of Multi-linear map
Multi-linear map is used to define the determinant. However, since the multi-linear map itself does not have linearity, I feel a sense of heterogeneity when I compare this concept with the contents ...
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Why is Einstein summation named after Einstein?
Why is Einstein summation of tensors (summation of repeated indices) named after Einstein?
"Einstein rule" in the Encyclopedia of Mathematics only says:
This rule was proposed by A. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $P$...
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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"....
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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 $S_n$.
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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 ...
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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 ...
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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?...
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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 ...
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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 $...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Jordan's Paper on the Jordan Canonical Form
In which paper, did Jordan introduce/prove the Jordan canonical form?
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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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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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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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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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