There exists many methods for multiply a square matrix A by itself m times very quickly and efficiently. However, I am need to do the multiplication $(A+X_1)(A+X_2)(A+X_3)...(A+X_m)$ where $m$ is large, and the $X_i$ are matrices.
Is there an easy way to to this that is less intensive, using the fact that the result of $A^m$ is known? (All the $X_i$ have the same form, where non-zero values are in the same positions in the matrix, but their values for each $X_i$ differ. Also, if it helps towards reducing the number of operations, the trace of this product needs to be taken).
Many thanks