  We calculated reduced density characteristic matrix (RDCM) for quantum walk
on cycles (QWC) to study asymptotic properties of the most general form of
quantum walk on cycles with general $U(2)$ coin operator. As an example,
entanglement temperature for general initial state has been calculated and
compared to previous results. Also, we have modified RDCM to derive analytical
expression for general form of limiting distribution (LD).
