The paper presents a continuation of authors’ research on linear periodic discrete-time systems \[ x(k+1)=A(k)x(k)+B(k)u(k),\quad u(k)=F(k)x(k)+u_ 0(k), \] where \(A(k),B(k)\) and \(F(k)\) are periodic matrices with fixed period [see the authors, Int. J. Control, 43, 517-537 (1986; Zbl 0583.93044), ibid., 45, 1603-1626 (1987; Zbl 0622.93042), Automatica 24, No. 3, 375-385 (1988; Zbl 0653.93033)]. It is proved that if the system is not reachable, it is possible to assign a spectrum minus a feedback independent zero eigenvalued with time-varying multiplicity for the reachable part over a period.