All Questions
Tagged with linear-algebra nt.number-theory
215 questions
0
votes
0
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118
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Uncomplete argument in Nishioka book
In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
- 3,998
1
vote
3
answers
257
views
Eigenvalues of positive matrices in $\mathrm{SL}(d,\mathbb{Z})$
Let $A\in\operatorname{SL}(d,\mathbb{Z})$ be an irreducible positive matrix, i. e. $A=(a_{i,j})_{1\leq i,j\leq d}$ with $a_{i,j}\in\mathbb{Z}_{>0}$. From the Perron-Frobenius theorem, we know that $...
- 11
1
vote
0
answers
67
views
System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
- 25.4k
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
21
votes
0
answers
520
views
Is the exponent of $2$ in the Pythagorean theorem the "same $2$" as $[\mathbb{C} : \mathbb{R}]$?
I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...
- 1,206
2
votes
0
answers
108
views
Largest prime determinant of a binary matrix
Given an integer $n$, I want to prove the existence of an $n\times n$ binary matrix (with 0,1 entries), whose determinant is a prime number. What is a lower bound on the largest determinant that I ...
- 3,549
5
votes
1
answer
303
views
Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 696
1
vote
0
answers
195
views
Conjectural values of some determinants involving Legendre symbols (II)
Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by the evaluation of the determinants
$$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{...
- 15.6k
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
4
votes
0
answers
238
views
Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
2
votes
1
answer
99
views
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 ...
- 23
0
votes
2
answers
252
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 $x_{i} \geq 0$ for all $i=1,\...
- 227
0
votes
0
answers
163
views
Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
2
votes
0
answers
112
views
Invariant factors and commuting matrices over a discrete valuation ring
$\DeclareMathOperator\Im{Im}\DeclareMathOperator\Ker{Ker}$Let $A$ be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T$, $Y^T$ be ...
2
votes
1
answer
226
views
Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?
I am trying to prove that the function:
$$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$
is a positive definite function over the natural numbers. What has sometimes ...
- 3,064
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
7
votes
1
answer
633
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
1
vote
1
answer
252
views
Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 823
4
votes
1
answer
183
views
What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?
Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
- 143
1
vote
0
answers
158
views
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
18
votes
2
answers
488
views
Encoding primes via ranks of sign matrices
(Reposted from math.SE)
Recently I came across a very simply defined family of matrices: for $n \in \mathbb{N}$, set $A_n := (a_{ij})_{0 \le i, j \le n-1}$, where
$$\displaystyle a_{ij} := (-1)^{\big\...
- 281
8
votes
2
answers
528
views
Number of matrices with unit determinant and fixed sum of elements
Question. Let $\mathcal{M}_3$ be the set of $3\times 3$ matrices with non-negative integer entries and unit determinant. What is the number of $M\in \mathcal{M}_3$ with fixed sum of entries? What is ...
- 577
2
votes
2
answers
310
views
Factoring positive semidefinite matrices over $\mathbb{Q}[i]$
Let $P\in \mathrm{GL}_n(\mathbb{Q})$ be positive definite, and $(\det P)^{\frac{1}{n}}\in \mathbb{Q}$.
Question. Can one always write $P=(\det P)^{\frac{1}{n}} AA^*$, with $A\in \mathrm{SL}_n(\mathbb{...
- 14k
0
votes
0
answers
117
views
Covering zeroes of quadratic forms by linear forms
Consider a quadratic form
$$Q(x_1,\ldots,x_n)=\sum_{i,j}a_{ij}x_ix_j,$$
where $a_{i,j}\in \mathbb{R}$ and $x_i\in A$ for some $A\subset \mathbb{R}$ such what $|A|=k$.
Question. What is the smallest ...
- 2,288
4
votes
1
answer
177
views
Density of points in the torus whose iterates under a matrix converge to zero
In Yves Benoist and Jean-François Quint's notes Introduction to random walks on homogeneous spaces (top of page 11),
the following is listed as a step in the non-Fourier analytic proof of ergodicity ...
- 481
1
vote
1
answer
296
views
A query about modular arithmetic on a matrix
Given a matrix $M$ that consists of a set of $4K$ binary row vectors (each vector entry is 0 or 1) each of length $K$. Moreover, it is known/promised that no subset of rows in matrix add to an all 1 ...
- 13
5
votes
0
answers
196
views
Is it true that the $\mathbb{F}_p$-rank of a linear combination of matrices is usually not smaller than its $\mathbb{Q}$-rank?
Consider fixed $3 \times 3$ integer matrices $A_1,A_2,A_3$ and the $\sim H^3$ linear combination matrices $A(\mathbf{h})=h_1A_1+h_2A_2+h_3A_3$ where $h_1,h_2,h_3$ are integers with $\vert h_i\vert \le ...
0
votes
1
answer
269
views
Product of subspace and its inverse
$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
2
votes
2
answers
208
views
Successive minima of a lattice and projection along the the shortest nonzero vector
Let $\mathcal L$ be the space of lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...
- 457
1
vote
2
answers
204
views
Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs
This might be related to counting hamiltonian cycles.
@Peter Taylor gave negative result about the one dimensional case, but we believe his attack is
not directly applicable to this question.
Given ...
- 25.4k
6
votes
2
answers
339
views
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
6
votes
1
answer
402
views
Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$
For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not.
...
- 15.6k
2
votes
4
answers
751
views
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
How to prove without using advanced theorems that quaternions algebra $H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that $H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$?
My ideas: I ...
0
votes
0
answers
151
views
CRT for linear forms
Suppose $A,B,C,A',B',C'$ are random distinct primes in $[T,2T]$ and $u,v$ are integers in $[T,2T]$.
Suppose we know:
$$Au+Bv\equiv r\bmod C$$
$$A'u+B'v\equiv r'\bmod C'$$
can we identify $u,v$ in ...
- 13.9k
2
votes
1
answer
389
views
Existence of regular semisimple elements in linear group over local field
Let $ L $ be a finite extension of $p$-adic numbers $ \mathbb{Q}_p $. Let $ \text{GL}_{n}(L) $ denote the general linear group $ \text{GL}_{n}(L) $ over $L$ equipped with the topology induced from the ...
- 863
3
votes
0
answers
77
views
Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
4
votes
0
answers
211
views
Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
25
votes
2
answers
872
views
Show that these matrices are invertible for all $p>3$
I am working on a paper which will extend a result in my thesis and have boiled one problem down to the following: show that the symmetric matrix $M_p$, whose definition follows, is invertible for all ...
- 261
4
votes
1
answer
239
views
Yet, another numerical variant of the Vandermonde matrix
In my earlier (soft) MO post, an elementary response was given by Ofir Gorodetsky in regard to the determinant of the symbolic counterpart to the numerical matrix $\mathbf{M}_n=(i^j-j^i)_{i,j}^{1,n}$.
...
- 43.2k
1
vote
0
answers
125
views
Determinants associated with Stern's diatomic sequence
Consider the so-called Stern's triangle (refer to these slides by R. Stanley), we denote here by $a_n(k)$. In an article Some linear recurrences motivated by Stern’s diatomic array, Stanley provided ...
- 43.2k
4
votes
2
answers
522
views
How far can the $\mathbb{F}_p$-rank of an integer matrix with small entries drop?
Let $n$ and $k$ be some fixed integers with $1 \le k \le n$.
I am interested in conditions on the size of the integer coefficients of a non-singular $n \times n$-matrix $A$ which ensure that the $\...
0
votes
1
answer
125
views
Special type of normal form of matrix in principal ideal domain
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric.
Can one always ...
- 145
0
votes
1
answer
199
views
Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
- 923
1
vote
1
answer
144
views
On parametrization of a type of unimodular $2\times2$ integral matrix
A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds.
Is there a parametrization of such matrices with $|w||z|-xy=1$
$$w,z<0<\max(...
- 13.9k
2
votes
1
answer
193
views
Irreducible components of a cyclic extension over $ \mathbb{Q} $
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then ...
- 923
10
votes
1
answer
816
views
$\text{SL}_2(\mathbb{Z})$ and continued fractions?
I know the following facts: $\text{SL}_2(\mathbb{Z})$ is generated by everyone's favorite matrices
\begin{equation*}
S =
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\end{equation*}
and
\...
- 1,075
1
vote
0
answers
93
views
Conjectures about the automorphism group of integer lattice by enlarging the matrix
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Notation: $\GL(n, \mathbb{Z})$ to be the set of $n \times n$ invertible matrix, and ...
- 145
1
vote
0
answers
91
views
Diophantine equation about the automorphism group of lattice by constraints
Fixed $\sigma_x=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)$ and $K=\left(
\begin{array}{ccc}
3 & 32 & -64 \\
1 & 32 & -32 \\
-2 & -32 & 64 \\
\...
- 149
4
votes
0
answers
113
views
Index of norm $ 1 $ subgroup in a cyclic extension
Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$...
- 923
1
vote
0
answers
136
views
Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $6$
Let us consider a primitive $7^{\text{th}}$ root of unity $\eta$. Then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6}$. So the dimension of the $\mathbb{Q}$-...
- 923