  A necessary and sufficient conditions for certain class of periodic unitary
transition operators to have eigenvalues are given. Applying this, it is shown
that Grover walks in any dimension has both of $\pm 1$ as eigenvalues and it
has no other eigenvalues. It is also shown that the lazy Grover walks in any
dimension has $1$ as an eigenvalue, and it has no other eigenvalues. As a
result, a localization phenomenon occurs for these quantum walks. A general
criterion for the existence of eigenvalues can be applied also to certain
quantum walks of Fourier type. It is shown that the two-dimensional Fourier
walk does not have eigenvalues and hence it is not localized at any point. Some
other topics such as Grover walks on the triangular lattice, products and
deformations of Grover walks are also discussed.
