The multinomial theorem asserts that$$(x_1+\cdots + x_k)^n = \sum_{n_1+\cdots +n_k = n,\ n_1,\dots,n_k\geq0} \frac{n!}{n_1! \cdots n_k!}x_1^{n_1}\cdots x_k^{n_l}.$$How does this fomrula change when we replace $x_1,\dots, x_k$ with matrices $A_1,\dots, A_k$? That is, how would one express $(A_1+\cdots A_k)^m$ as above? I suspect now that the order of multiplication matters, one would have to rather sum over all products of $A_1,\dots,A_k$ of length $n$. Would this be correct, and if so, how would one write out the formula?
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