Euclidean lattices with a metric
Let $L_1, L_2 \subseteq \mathbb{R}^n$ be full-rank Euclidean lattices with generating matrices $A, B$ respectively, and let $d(A,B)< \epsilon$, for some fixed positive real $\epsilon$. Then do there exist $\gamma, \delta \in \mathbb{R}^{+}$ (depending on $A,B, \epsilon$) such that \[\gamma < \operatorname{sup}_{x \in \mathbb{R}^n} |Ax-Bx|< \delta?\]
One clue could be that $\gamma, \delta$ somehow depend on the smallest and largest (in absolute value) eigenvalues of $A,B$.
Definition of $d(A,B)$: Given (full rank) lattices $L_1 = (a_{ij}), L_2=(b_{ij}) \in \mathbb{R}^n$ with gen. matrices $A, B$ respectively, we define \[d(L_1,L_2)=\sqrt{\sum_{i=1}^n \sum_{j=1}^n}(a_{ij}-b_{ij})^2.\]
98
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1.6K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 172 views
- $30.00
Related Questions
- ALGEBRA WORD PROBLEM - Trajectory of a NASA rocket
- Multiplying inequalities by x or ^2
- Double absolute value equations.
- Prove that $A - B=A\cap B^c$
- College Algebra Help
- Stuck on this and need the answer for this problem at 6. Thanks
- Absolute value functions.
- How do you go about solving this question?