标题： 保持秩的索引相关矩阵变换：时钟机制和分解理论空间模型的应用 显示英文标题

标题： Rank-Preserving Index-Dependent Matrix Transformations: Applications to Clockwork and Deconstruction Theory Space Models

摘要： 我们引入了一个依赖于索引的元素级矩阵变换的通用框架，$b_{ij} = a_{ij} / g_f(i,j)$，它在高能物理中生成质量层次结构方面有直接应用。 本文建立了对$g_f(i,j)$的精确数学条件，以保持原始矩阵的秩和零空间。 我们的研究发现，这种变换为工程矩阵的零空间特定属性提供了一种强大方法；通过适当选择函数$g_f(i,j)$，可以生成具有多样且可控局域化模式的零向量（或特征向量）。 讨论了该技术的广泛应用，给出了来自高能物理的详细示例。 我们展示了如何使用我们的框架来定制钟表模型和维度分解模型中的0模轮廓和费米子质量谱，表明标准钟表机制作为$(g_f(i,j) = f^{(i-j)})$的一个特例出现，从而为粒子物理超出标准模型（BSM）的模型构建提供了新工具。 这项工作展示了这些变换在各种领域模型构建中的潜力，其中局域模式或特定光谱特性是关键。 显示英文摘要

摘要： We introduce a versatile framework of index-dependent element-wise matrix transformations, $b_{ij} = a_{ij} / g_f(i,j)$, with direct applications to hierarchy generating mass hierarchies in high-energy physics. This paper establishes the precise mathematical conditions on $g_f(i,j)$ that preserve the rank and nullity of the original matrix. Our study reveals that such transformations provide a powerful method for engineering specific properties of a matrix's null space; by appropriately selecting the function $g_f(i,j)$, one can generate null vectors (or eigenvectors) with diverse and controllable localization patterns. The broad applicability of this technique is discussed, with detailed examples drawn from high-energy physics. We demonstrate how our framework can be used to tailor 0-mode profiles and fermionic mass spectra in clockwork and dimensional deconstruction models, showing that the standard clockwork mechanism arises as a particular case $(g_f(i,j) = f^{(i-j)})$, thereby offering new tools for particle physics BSM model building. This work illustrates the potential of these transformations in model building across various fields where localized modes or specific spectral properties are crucial.