In this paper, we consider the homogeneous difference equation \[ \sum _{j = 0}^k {\alpha _j y_{n - j} } = 0,\quad n = k,k + 1,k + 2, \cdots ,\] with initial values \[ y_j = q_j,\quad j = 0(1)k - 1 .\] The $y_j$ are d-component column vectors, the $\alpha _j $ are $d \times d$ matrices independent of n. We derive algebraic criteria for numerical stability of the difference equation, which is understood in the sense that the solution $\{ y_j \} $ and its difference quotients up to order $s \in \{ {0,1,2,3, \ldots } \}$ depend continuously on the initial values $\{ q_j \} $. This generalizes the well-known case where $s = 0$ and the $\alpha _j $ are diagonal matrices.