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

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.