All Questions
9 questions
16
votes
4
answers
930
views
Integer matrices whose determinant equals their norm
Let $M$ be an $2 \times 2$ matrix, with all entries in $\mathbb{N}$:
$$
M=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \;.
$$
So
$$
\mathrm{det}(M) = a d - b c \; .
$$
The
Euclidean norm
(...
- 151k
8
votes
2
answers
2k
views
Algorithm for solving systems of linear Diophantine inequalities
So, I posted on StackOverflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...
- 3,079
5
votes
0
answers
743
views
Is this set empty?
Suppose we have two rank $n-1$ matrices in $\Bbb Z^{(n-1)\times n}$ given by
$$C=\begin{bmatrix}
c_{1}& -d_{1}& 0& 0&\dots 0& 0\\
0& c_{2}& -d_{2}& 0&...
- 13.9k
1
vote
1
answer
144
views
On parametrization of a type of unimodular $2\times2$ integral matrix
A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds.
Is there a parametrization of such matrices with $|w||z|-xy=1$
$$w,z<0<\max(...
- 13.9k
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
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
0
votes
1
answer
637
views
Rational solutions of homogeneous equations
Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\...
- 10.4k
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