  Mathematical analysis on the existence of eigenvalues is vital, as it
corresponds to the occurrence of localization, an exceptionally important
property of quantum walks. Previous studies have demonstrated that eigenvalue
analysis utilizing the transfer matrix proves beneficial for space
inhomogeneous three-state quantum walks with a specific class of coin matrices,
including Grover matrices. In this research, we turn our attention to the
transfer matrix of three-state quantum walks with a general coin matrix.
Building upon previous research methodologies, we dive deeper into
investigating the properties of the transfer matrix and employ numerical
analysis to derive eigenvalues for models that were previously unanalyzable.
