Suppose that $\mathbf A$ and $\mathbf B$ are square, diagonalizable matrices. Consider the following infinite sum of all combinations of these two matrices: \begin{align} \mathbf S = \mathbf I &+\mathbf A + \mathbf B +\\ &+\mathbf A^2 + \mathbf A\mathbf B + \mathbf B\mathbf A + \mathbf B^2+ \\ &+\mathbf A^3 + \mathbf A^2\mathbf B + \mathbf A\mathbf B\mathbf A+ \mathbf A\mathbf B^2 + \mathbf B\mathbf A^2+ \mathbf B\mathbf A\mathbf B+ \mathbf B^2\mathbf A + \mathbf B^3+\\ &+\cdots \end{align} Let $\mathbf A=\mathbf Q_A\mathbf \Lambda_A\mathbf Q_A^{-1}$, and $\mathbf B=\mathbf Q_B\mathbf \Lambda_B\mathbf Q_B^{-1}$ be the eigendecompositions of $\mathbf A$ and $\mathbf B$.
If $\mathbf B=\mathbf 0$, then the sum would be equal to $$\mathbf S = \sum_{k=0}^\infty \mathbf A^k=\sum_{k=0}^\infty(\mathbf Q_A\mathbf \Lambda_A\mathbf Q_A^{-1})^k=\mathbf Q_A \left(\sum_{k=0}^\infty\mathbf\Lambda_A^k\right)\mathbf Q_A^{-1}.$$
Is there a general way to express $\mathbf S$ as a function of $\mathbf Q_A$, $\mathbf Q_B$, and sums that involve $\mathbf \Lambda_A$ and $\mathbf \Lambda_B$?