结构随机矩阵的规范
Radosław Adamczak1, Joscha Prochno2, Marta Strzelecka1,3
1Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland.
Mathematische annalen
|August 12, 2024
概括
本研究分析了结构随机矩阵的预期运算符规范. 我们为各种随机矩阵类型建立了最佳边界,在特定情况下提供了精确的顺序确定.
科学领域:
- 随机矩阵理论 随机矩阵理论
- 功能分析是一种功能分析.
- 运算子理论 运算子理论
背景情况:
- 随机矩阵的研究在各种科学领域至关重要.
- 了解结构随机矩阵的运算符规范是一个复杂的问题.
- 现有的研究往往侧重于特定类型的随机矩阵或规范.
研究的目的:
- 为了研究结构化随机矩阵的预期运算符规范.
- 为了在不同的随机矩阵分布中建立这些规范的最佳界限.
- 在特定场景中确定预期规范的精确顺序.
主要方法:
- 分析结构随机矩阵 $X_A$ 作为 $\ell_p^n$ 和 $\ell_q^m$ 空间之间的运算符.
- 对于具有i.i.d.矩阵的预期运算符规范的边界推导. 高斯式,独立的平均值为零的边界,以及独立的平均值为零的 $\psi_r$ 条目.
- 使用运算符规范表达结果的确定性矩阵$A$的哈达马德积和它的转换.
主要成果:
- 在随机矩阵$X$的条目上的各种假设下,为预期的运算符规范证明了最佳边界 (高达对数项).
- 在某些情况下,预期规范的精确顺序被确定为常量.
- 这些发现将$X_A$的运算子规范与通过哈达马德积的确定性矩阵$A$的属性联系起来.
结论:
- 这项研究在理解结构随机矩阵的运算符规范方面取得了重大进展.
- 导出的边界和精确的顺序结果为理论和应用研究提供了宝贵的见解.
- 该方法和发现适用于一系列涉及随机矩阵的数学和统计问题.
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