We obtain some oscillation criteria for the Hamiltonian difference system
\[
{
Δ
Y
(
t
)
=
B
(
t
)
Y
(
t
+
1
)
+
C
(
t
)
Z
(
t
)
,
Δ
Z
(
t
)
=
−
A
(
t
)
Y
(
t
+
1
)
−
B
∗
(
t
)
Z
(
t
)
,
\left \{ \begin {gathered} \Delta Y(t) = B(t)Y(t + 1) + C(t)Z(t), \hfill \\ \Delta Z(t) = - A(t)Y(t + 1) - {B^{\ast }}(t)Z(t), \hfill \\ \end {gathered} \right .
\]
where
A
,
B
,
C
,
Y
,
Z
A,B,C,Y,Z
are
d
×
d
d \times d
matrix functions. As a corollary, we establish the validity of an earlier conjecture for a second-order matrix difference system.