I have a matrix $Q := I_{2m} - t A$, where
\begin{equation*} A = \frac{1}{2} \begin{bmatrix} GC^{-1}(GC^{-1})^T & -GC^{-1}(GC^{-1})^T \\ -GC^{-1}(GC^{-1})^T & GC^{-1}(GC^{-1})^T \\ \end{bmatrix}, \end{equation*}
$G \in \mathbb{R}^{m\times n}$ is a sparse matrix, $C \in \mathbb{R}^{n\times n}$ is a symmetric circulant matrix and $t \in \mathbb{R}^+$. I want to find the most efficient way to numerically calculate $Q^k$
I know an efficient way to do this is to diagonalize $Q$ using eigenvectors. However, since $C$ is circulant and $G$ is sparse, it makes me wonder if there is a way I can use FFT's to make this even faster.
For an easier version of the problem: Set $C$ to identity.