The paper explores the dynamics of quantum systems using Krylov space methods, focusing on tridiagonal matrices and their behavior across different phases. The Rosenzweig-Porter (RP) model is used as a primary example, which exhibits a fractal regime in addition to ergodic and localized phases. The authors analyze the properties of tridiagonal matrix elements and basis vectors in random matrix ensembles, particularly in the RP model. They introduce tools to identify transition points between these phases and provide exact expressions for Lanczos coefficients in terms of $q$-logarithmic functions. Numerical results are supported by analytical reasoning, and the complexity of Krylov states within these phases is investigated. The study reveals the efficacy of Krylov space methods in pinpointing transitions and provides insights into the behavior of Lanczos coefficients and the Inverse Participation Ratio (IPR) in different phases. The paper also discusses the spread complexity of Krylov states and its behavior across the ergodic, fractal, and localized regimes.The paper explores the dynamics of quantum systems using Krylov space methods, focusing on tridiagonal matrices and their behavior across different phases. The Rosenzweig-Porter (RP) model is used as a primary example, which exhibits a fractal regime in addition to ergodic and localized phases. The authors analyze the properties of tridiagonal matrix elements and basis vectors in random matrix ensembles, particularly in the RP model. They introduce tools to identify transition points between these phases and provide exact expressions for Lanczos coefficients in terms of $q$-logarithmic functions. Numerical results are supported by analytical reasoning, and the complexity of Krylov states within these phases is investigated. The study reveals the efficacy of Krylov space methods in pinpointing transitions and provides insights into the behavior of Lanczos coefficients and the Inverse Participation Ratio (IPR) in different phases. The paper also discusses the spread complexity of Krylov states and its behavior across the ergodic, fractal, and localized regimes.