This study investigates the Krylov fractality and complexity in the Rosenzweig-Porter (RP) model, a random matrix ensemble that exhibits extended, fractal, and localized phases. The research employs Krylov space techniques to analyze the dynamical aspects of the model, focusing on the behavior of tridiagonal matrices and their associated Lanczos coefficients. The RP model is characterized by a parameter γ that controls the transition between ergodic, fractal, and localized phases. The study provides exact expressions for Lanczos coefficients in terms of q-logarithmic functions and analyzes the Krylov spectrum, showing how it transitions across these phases. The Krylov Inverse Participation Ratio (IPR) and fractal dimension are introduced to identify critical transition points. The results demonstrate that the Krylov state complexity, measured by spread complexity, exhibits distinct behaviors in the ergodic, fractal, and localized regimes, with pronounced peaks in the former two. The study also shows that the Krylov IPR scales differently in each phase, reflecting the degree of localization. The findings highlight the effectiveness of Krylov space methods in detecting phase transitions and provide a framework for understanding the dynamics of quantum systems in random matrix ensembles. The results contribute to the broader understanding of Krylov techniques in physics and offer a new perspective on the behavior of complex systems.This study investigates the Krylov fractality and complexity in the Rosenzweig-Porter (RP) model, a random matrix ensemble that exhibits extended, fractal, and localized phases. The research employs Krylov space techniques to analyze the dynamical aspects of the model, focusing on the behavior of tridiagonal matrices and their associated Lanczos coefficients. The RP model is characterized by a parameter γ that controls the transition between ergodic, fractal, and localized phases. The study provides exact expressions for Lanczos coefficients in terms of q-logarithmic functions and analyzes the Krylov spectrum, showing how it transitions across these phases. The Krylov Inverse Participation Ratio (IPR) and fractal dimension are introduced to identify critical transition points. The results demonstrate that the Krylov state complexity, measured by spread complexity, exhibits distinct behaviors in the ergodic, fractal, and localized regimes, with pronounced peaks in the former two. The study also shows that the Krylov IPR scales differently in each phase, reflecting the degree of localization. The findings highlight the effectiveness of Krylov space methods in detecting phase transitions and provide a framework for understanding the dynamics of quantum systems in random matrix ensembles. The results contribute to the broader understanding of Krylov techniques in physics and offer a new perspective on the behavior of complex systems.