Power of a block matrix with eigenvalues on the unit circle In the expression
$$\begin{bmatrix}A & C \\ 0 & B\end{bmatrix}^n = \begin{bmatrix}A^n & * \\ 0 & B^n\end{bmatrix},$$
I wonder whether the term denoted by * can be expressed in a simple form when we assume the following: (1) $A$ has its eigenvalues on or inside the unit circle. Those on the
        unit circle are simple;  (2)  $B$ has its eigenvalues strictly inside the unit circle; (3)  $A$ and $B$ may have different dimensions.
In fact, I am interested for the value of * as $n \rightarrow \infty$. It would be $C(I-B)^{-1}$ when $A=I$ but in a general case, $A^n$, though bounded, may not converge as $n \rightarrow \infty$. So, I wonder whether * can have a simple expression.
 A: Let
$$M (n) := \left[\begin{array}{cc} A & C\\ 0 & B\end{array}\right]^n = \left[\begin{array}{cc} A^n & U (n)\\ 0 & B^n\end{array}\right]$$
where I call $U (n)$ what you call $*$. Thus, we have that $M (n+1)$ is given by
$$M (n+1) = \left[\begin{array}{cc} A^{n+1} & U (n+1)\\ 0 & B^{n+1}\end{array}\right] = \left[\begin{array}{cc} A & C\\ 0 & B\end{array}\right] \left[\begin{array}{cc} A^n & U (n)\\ 0 & B^n\end{array}\right]$$
and, therefore, we obtain the following matrix difference equation
$$U (n+1) = A \, U (n) + C \, B^n$$
Note that $U (0) = 0$, $U (1) = C$, and $U (2) = A \, C + C \, B$. The value of $U (n)$ as $n$ goes to infinity, which we denote by $\bar{U}$, is given by $\bar{U} = A \, \bar{U} + C B^{\infty}$, which yields $\bar{U} = (I - A)^{-1} C B^{\infty}$. Since you say that $B$ "has its eigenvalues strictly inside the unit circle", we conclude that $B^{\infty} = 0$ and, hence, $\bar{U} = 0$. One can also easily obtain the general solution, which is
$$U (n) = A^n \, U (0) + \displaystyle \sum_{i=0}^{n-1} A^i \, C \, B^{n-1-i}$$
and since $U (0) = 0$, the "natural response" is zero and we are left with the "forced response"
$$U (n) = \displaystyle \sum_{i=0}^{n-1} A^i \, C \, B^{n-1-i}$$
