  There is a property called localization, which is essential for applications
of quantum walks. From a mathematical point of view, the occurrence of
localization is known to be equivalent to the existence of eigenvalues of the
time evolution operators, which are defined by coin matrices. A previous study
proposed an approach to the eigenvalue problem for space-inhomogeneous models
using transfer matrices. However, the approach was restricted to models whose
coin matrices are the same in positions sufficiently far to the left and right,
respectively. This study shows that the method can be applied to extended
models with periodically arranged coin matrices. Moreover, we investigate
localization by performing the eigenvalue analysis and deriving their
time-averaged limit distribution.
