All Questions
Tagged with linear-algebra reference-request
318 questions
1
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1
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Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
3
votes
1
answer
327
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Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 43.2k
7
votes
4
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558
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Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 12.7k
3
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1
answer
232
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Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
4
votes
1
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170
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About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 696
4
votes
0
answers
284
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Institutional approach to linear algebra
In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following
Definition. An institution ...
- 10.1k
4
votes
1
answer
686
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Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 41.9k
6
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0
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111
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Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
1
vote
0
answers
64
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Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
- 11
6
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1
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244
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Linear independence over field of rational functions
To prove that functions $f_1(x), \dots, f_n(x)$ with $x \in \mathbb R$ are linearly independent, we only need to show that the Wronskian of these functions is non-zero at a certain value of $x$. Now ...
- 1,608
6
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1
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239
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Attempts to define a matrix exponential over (as much as possible) general fields
Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...
- 361
1
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0
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29
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Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?
Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
- 11
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1
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103
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Interpolation between two matrices so that $L^p$ norm is controlled
Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
- 697
0
votes
1
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525
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What is the mathematician's definition of the determinant? [closed]
I am trying really hard to find a good definition of the determinant.
I have looked virtually every single resource online and everybody gives a different answer:
sum of cofactors or minors https://...
- 179
1
vote
0
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61
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Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues
I am looking for a reference justifying the following statement.
Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$
$$
...
- 5,380
6
votes
1
answer
1k
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Full expansion of $\det(I+\varepsilon A)$
It is well known that given a $n \times n$ matrix $A$, it holds that
$$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$
I would need a full representation of $ \...
- 697
5
votes
0
answers
202
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Difficulty of solving $Ax=b$ in terms of limiting spectral density of $A$?
Suppose $A$ is a random real-valued $n\times n$ matrix and we want to know the difficulty of solving $Ax=b$ when entries of $b$ are sampled IID from Normal$(0,1)$.
Can we say anything about the ...
- 2,252
2
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0
answers
269
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Singular values of Kronecker product of random matrices
I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
- 121
5
votes
1
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141
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On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
- 696
8
votes
1
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324
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On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
- 128k
1
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0
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158
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Hankel transform of certain $\pm1$ sequences
The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$
where $s_2(k)$ is ...
- 43.2k
5
votes
2
answers
318
views
An inequality problem for certain positive-definite matrices
Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $<0$. Let $a$ be a column $n\times1$ matrix such ...
- 128k
6
votes
1
answer
217
views
An inequality for certain positive-definite matrices
Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with ...
- 128k
5
votes
1
answer
474
views
An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 128k
5
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0
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190
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Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
4
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0
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181
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Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
12
votes
3
answers
784
views
Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 128k
14
votes
1
answer
417
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Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
2
votes
1
answer
358
views
q-polynomials in terms of a basis
Consider the polynomials
$$f_n(q)=\prod_{j=1}^n(1+q^j) \qquad \text{and} \qquad g_m(q)=1+q+q^2+\cdots+q^m.$$
I'll list a few examples to motivate my question. Direct calculations show that
$$f_1=g_1, \...
- 43.2k
0
votes
0
answers
44
views
Sufficient conditions to order the solutions to a system of linear equations
A pretty elementary question, but does anyone know of sufficient conditions to order the solutions of a system of linear equations? For example, in the system, \begin{align*}\begin{bmatrix}a_{11}&...
- 1
2
votes
1
answer
223
views
Linearly independent vectors from a family of subspaces
For the proof of a certain combinatorial statement on subsets of ${\mathbb F}_2^n$
in a paper I and several other people are working on, the following statement was helpful.
Proposition.
Let $V$ be a ...
- 11.3k
5
votes
2
answers
276
views
Dilation of bounded linear operators
Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
- 51
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0
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198
views
On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
1
vote
0
answers
61
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Homology of infinite matroids of finite rank
Bjorner has a great paper about the homology of independence complexes of finite matroids, which is the usual context in matroid theory as far as I understand. However, I've also been told that often ...
- 2,044
0
votes
0
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145
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Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
4
votes
0
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112
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.2k
1
vote
2
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333
views
Condition for equality of modules generated by columns of matrices
Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
- 431
3
votes
1
answer
203
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Results on Boolean matrices
Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
- 31
7
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0
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195
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Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.6k
6
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2
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455
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An elementary inequality of operators
Suppose $a,b$ are two positive-definite linear operators on (say) $\mathbb R^n$. For $p\in(0,1)$, do we then have $(a+b)^p\leq a^p+b^p$ (with respect to the Loewner order)?
- 61
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0
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233
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Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
0
votes
0
answers
78
views
Topology of independence set of a vector space
This seems like something that would have a well-known treatment somewhere, but I'm not sure where to look. If we have a vector space $V$ (or maybe even a module), we can consider an abstract ...
- 2,044
17
votes
1
answer
668
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A coincidence or a fact: determinants of two matrices
While playing around with the MO question Determinant with factorials is not 0? about a determinant of the Hankel matrix of entries $(i+j-2)!$, having the value $\prod_{k=0}^{n-1}k!^2$, I stumbled on ...
- 43.2k
3
votes
1
answer
175
views
Is there a classical textbook/reference on numerical discretization schemes?
I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-...
- 479
0
votes
1
answer
129
views
A variant of numeric Vandermonde which failed symbolically?
Given some variables $x_1, x_2, \dots, x_n$, the Vandermonde determinant is given by
$$V_n(x_1,\dots,x_n):=\det(x_j^{i-1})_{i,j=1}^n=\prod_{i<j}(x_j-x_i).$$
One can take as special cases: $x_j=j$ ...
- 43.2k
3
votes
0
answers
185
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"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.2k
6
votes
2
answers
339
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Sum of divisors and LCM in determinants
$\newcommand{\lcm}{\operatorname{lcm}}$Let $\gcd(i,j)$ and $\lcm(i,j)$ be the greatest common divisor and least common multiple of the pair of positive integers $i$ and $j$. Denote the sum of divisors ...
- 43.2k
2
votes
1
answer
460
views
Maximum permuted row/column sum of a matrix
Given a real $n \times n$ matrix $A$ (feel free to assume its entries are non-negative, if it helps), what is known about the problem of computing the quantity
$$
\max_{\sigma \in S_n} \left\{\sum_{j=...
- 6,351
20
votes
2
answers
1k
views
Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative
I saw this problem some years ago and I would greatly appreciate any reference or solution.
Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
- 3,153