  Krylov space methods provide an efficient framework for analyzing the
dynamical aspects of quantum systems, with tridiagonal matrices playing a key
role. Despite their importance, the behavior of such matrices from chaotic to
integrable states, transitioning through an intermediate phase, remains
unexplored. We aim to fill this gap by considering the properties of the
tridiagonal matrix elements and the associated basis vectors for appropriate
random matrix ensembles. We utilize the Rosenzweig-Porter model as our primary
example, which hosts a fractal regime in addition to the ergodic and localized
phases. We discuss the characteristics of the matrix elements and basis vectors
across the three (ergodic, fractal, and localized) regimes and introduce tools
to identify the transition points. The exact expressions of the Lanczos
coefficients are provided in terms of $q$-logarithmic function across the full
parameter regime. The numerical results are corroborated with analytical
reasoning for certain features of the Krylov spectra. Additionally, we
investigate the Krylov state complexity within these regimes, showcasing the
efficacy of our methods in pinpointing these transitions.
