For integer $k>0$, it is well-known that one can use binary exponentiation to evaluate the matrix power $\mathbf A^k$, where $\mathbf A$ is an $n\times n$ matrix.
However, it is not clear to me if that method can be readily modified to evaluate $\mathbf A^k \mathbf v$, where $\mathbf v$ is a vector of dimension $n$. Of course, one could just form the power as usual, and then apply it to the vector, but in cases like this (e.g. if the matrix is very large), one would wish to avoid forming a whole matrix. (At worst, one could use a few vectors of comparable size.)
Is there a good way to compute the action of a matrix power on a vector for integer exponents? My attempts to search the literature have come up empty, but perhaps I am not using the right keywords.
Solutions, or at least pointers to the literature, would be very much appreciated.