Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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Solving system of linear diophantine equations over the integers

In general, solving a system of linear diophantine equations over the integers is polynomial time solvable on the size of the coefficients of the equations. I am interested in an extension of this ...
user1868607's user avatar
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Convex sets via fixed point equations

I have an equation of the general form $$ X = S \cup T X $$ where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
rimu's user avatar
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3 votes
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Monotonicity of matrix conjugation

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under ...
António Borges Santos's user avatar
1 vote
1 answer
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Isn't every algebraic operad equipped with a trivial weight?

In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem): Let $P$ be a connected weight graded differential ...
groupoid's user avatar
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3 votes
1 answer
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Why is this polynomial factorizable? [closed]

I met a curious problem on factorizing a homogenerous polynomial of degree 9. Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$: \begin{align} &\quad\left| \begin{array}...
LichenSDU's user avatar
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How to find two sets of vectors which satisfy a set of matrix equations [migrated]

In my trial to solve a system of matrix equations, I wish to find two sets of non-zero vectors of $\mathbb{R}^3$ (which may be not unique) $\{ A_i \}$ and $\{ B_i \}$ where $i \in I$ (an index set, ...
rooted-and-grounded's user avatar
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About a circular variant of Vandermonde matrix

Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
lntk's user avatar
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Calculating an ellipsoid with the same distance sum from each point on it's surface to the two foci [migrated]

I'm a software engineer and look for the answer to a question which I havent found a solution for. So please excuse my bad mathematical terminology in this question. I want to generate an ellipsoid ...
Leon Archinger's user avatar
1 vote
2 answers
104 views

Property for bounding matrix exponential

Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
KatsanikJr's user avatar
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A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$ I'll briefly describe the problem. We let $...
fusheng's user avatar
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Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?

Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates? Playing around with the case $n = 2$, I’m pretty sure ...
Tim Campion's user avatar
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Low rank matrix in a subspace of matrices

Let $V\subset M_{m,n}(\mathbb{C})$ be a $n$-dimensional subspace ($m\geq 2$). Is there $X\in V$ such that $1\leq \operatorname{rank} X\leq m-1$?
user509119's user avatar
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A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below) I want to ...
It'sMe's user avatar
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On the irreducible submodules of adjoint representations $\text{ad}^{0}$

Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
stupid boy's user avatar
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1 answer
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Stabilizing conjugacy classes of integer matrices

$\DeclareMathOperator{\Conj}{Conj} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\id}{id} \newcommand\Z{\mathbb{Z}}$ For an $n \times n$ integer matrix $A \in \GL_n(\Z)$, let $\Conj(A)$ be the ...
Ingrid's user avatar
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some problem about the discrete of the first derivative operator

I am reading a paper (Parameter Choice Strategies for Multipenalty Regularization Massimo Fornasier, Valeriya Naumova, and Sergei V. Pereverzyev SIAM Journal on Numerical Analysis 2014 52:4, 1770-1794)...
bing's user avatar
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Concentration of bilinear forms

This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
Felix Goldberg's user avatar
1 vote
1 answer
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Van der Waerden conjecture and Alexandrov-Fenchel inequality

The Van der Waerden conjecture is a lower estimate of the permanent of a doubly stochastic matrix. In this article in Wikipedia it is stated that Egorychev's proof uses the Alexandrov-Fenchel ...
asv's user avatar
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3 votes
1 answer
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Solving a recursion for polynomials defined by a matrix product

Define the polynomial $p_n(X) \in \mathbb{Z}[X_1,...,X_n]$ as the top left entry in $A^n$ for the $(d \times d)$ matrix \begin{align*} & A = \left(\begin{matrix} X_1 & \dots & \...
Tardis's user avatar
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3 votes
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A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal

I apologize if this is too elementary a question, but I have not been able to make much progress. Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
user133281's user avatar
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2 answers
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Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix

Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
David Bevan's user avatar
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An isomorphic classification of non-associative division octonion algebras

A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...
Taras Banakh's user avatar
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Degree 6 Galois extension over $\mathbb{Q} $

Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
Sky's user avatar
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2 votes
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Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?

I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
BenJones's user avatar
3 votes
1 answer
234 views

Continuity of eigenvector of zero eigenvalue

Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
muddy's user avatar
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4 votes
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What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation \begin{align*} & X = A X A^T + \operatorname{Id} \tag{1} \...
Tardis's user avatar
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2 votes
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Perron-Frobenius theory for operators on matrices

Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (...
AdamNie's user avatar
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2 votes
1 answer
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Symmetric and anti-symmetric matrices and maximal eigenvalues

Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
Alapan Das's user avatar
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9 votes
3 answers
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$G$-module structure of the relation module for a presentation of a finite group $G$

Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
stupid_question_bot's user avatar
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1 answer
162 views

Reflections on subspaces of $\text{codim} > 1$

Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$. Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
Bipolar Minds's user avatar
1 vote
1 answer
101 views

Does a random matrix over $\mathbb{Z}_q$ map linearly independent vectors to statistically independent vectors?

Suppose $A \in \mathbb{Z}_q^{n \times m}$ is a random $n \times m$ matrix whose entries are i.i.d. uniform over $\mathbb{Z}_q=\mathbb{Z}/q\mathbb{Z}$, where $q\geq2$. Let $\mathbf{x}_1, \ldots, \...
aleph's user avatar
  • 503
0 votes
1 answer
140 views

Show that $\mathrm{PSL}_2(C)$ is complex algebraic [closed]

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\im{im}$I meet this problem when reading Artin's book Algebra. ...
Math Diego's user avatar
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1 answer
108 views

“Smallest” non-zero linear combination of vectors to obtain a non-negative vector

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix} where $x_{j} \geq 0$. Suppose that ...
Siddharth Iyer's user avatar
2 votes
1 answer
66 views

Lipschitz continuity of eigenprojections

This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
Guanaco96's user avatar
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0 answers
66 views

Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
pallab1234's user avatar
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0 answers
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Application of greedy approach for optimization

I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$ where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
Prakirt Raj's user avatar
3 votes
2 answers
157 views

Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$

Is there a closed-form solution for $$\max_D \operatorname{Tr}(ADBD)$$ where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...
CWC's user avatar
  • 351
1 vote
2 answers
137 views

Right inverse of integer matrix

If I have a rectangular matrix $A$ (say $4 \times 6$) with integer entries, is there a way to tell whether it has a right inverse that also has integer entries. I know that if $AA^T$ has determinant $...
user61388's user avatar
5 votes
0 answers
128 views

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
RandomTensor's user avatar
1 vote
1 answer
165 views

Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
loup blanc's user avatar
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0 votes
1 answer
109 views

Loss of degree for polynomials

Let $q$ be a power of a prime $p$, $m$ be an integer greater than $0$. Does there exist polynomials $P_0,\cdots,P_m$ of $\mathbb F_q[T]$ not all in $\mathbb F_q$ such that there exist polynomials $Q_0,...
joaopa's user avatar
  • 3,657
2 votes
0 answers
135 views

Spectrum of an almost Hamiltonian matrix

I have a complex-valued block matrix $N=\begin{bmatrix} A & B \\ C & -A^* \end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian. If $C$ were Hermitian, $N$ would ...
mathamphetamine's user avatar
0 votes
1 answer
67 views

Matrix quantization and effect on singular values

Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for $$ \| \sigma_i(A)-\...
ABIM's user avatar
  • 4,989
4 votes
1 answer
231 views

Does a generic linear map admit a vector whose iterates span $V$?

We say a linear map $T$ on a finite dimensional vector space $V$ admits spanning vectors if there exists some vector $v \in V$ whose iterates $v, Tv, T^2 v, \dots$ under $T$ span $V$. Question: ...
Nate River's user avatar
  • 4,802
0 votes
0 answers
103 views

Generalization of SVD algorithm

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
Martin Clever's user avatar
0 votes
1 answer
82 views

Find efficiently greatest difference between $2$ vectors from set of vectors [closed]

Let us have a list of vectors in a $3$D space. Is there a more efficient way to find the greatest difference between any two of them than combining each, computing the size of their difference, and ...
Honza S.'s user avatar
  • 109
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0 answers
25 views

The selection of minimal generating sets in Lie algebra

Suppose $A$ is a Lie algebra on field $F_{p}$ with $[A,A,A]=0$. Denote $\{a_{1},\cdots,a_{d}\}$ is a minimal generating set of $A$.It's possible that $[a_{i},a_{j}]=0$ for some $1\leq i<j\leq d$ ...
gdre's user avatar
  • 71
2 votes
0 answers
188 views

Is there a geometric or calculus-based reason why the following system of equations should have only one solution?

Let $x_1,x_2,x_3,x_4>0$. Consider the following cyclic system of equations: $$ 2 + x_2 + x_3 + x_4 + x_2 x_3 x_4 - 2 \left( \frac{x_2}{ \sqrt{x_1 x_2}} + \frac{x_3}{ \sqrt{x_1 x_3}} + \frac{x_4}{ \...
matilda's user avatar
  • 90
2 votes
0 answers
69 views

The rank of a certain linear combination of mutually commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be mutually commuting $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers. For any complex number $c$, let $A(c):=A_0+cA_1+c^2A_2+\ldots +c^rA_r$. We ...
sagnik chakraborty's user avatar
7 votes
2 answers
556 views

A curious equation on determinant----linear algebra or algebraic geometry?

I recently find a curious and unexplainable(as seems to me) equation on determinant as follows. $$3\begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ ...
LichenSDU's user avatar
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