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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

Peter Melech
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Hello
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  • Do you have more information on the particular form of the $X_i$? Are they upper/lower triangular for example? Do they commute with each other? Stuff like that could potentially be very helpful. – Mathematician 42 Sep 14 '18 at 13:04
  • There are no general form to the xi, as these depend on the system I need to analyse. They do not commute or anything as such. The only notable feature is that the first row is always zero – Hello Sep 14 '18 at 14:02
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    When A=0 you still have to multiply m matrices which takes about the same time as your original problem, since addition is faster than multiplication, so there can't be a way to speed it up significantly. – ericf Sep 14 '18 at 14:13

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