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I need to write in mathematica programming language a matrix Laurent Polynomial $$ T = \sum_{i=-5}^{5} A_i t ^i $$ where the $A_i$ are $n \times n$ matrices (I want the entries to be symbolic) which can take entries in $\mathbb{C}$. So that I can get the conditions on the entries of the $A_i$ such that $T^2$ and $T^3$ are just polynomials in $\mathbb{C}[t^2,t^3]$.

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    $\begingroup$ Have you seen Sum? $\endgroup$
    – Michael E2
    Sep 14 at 11:27
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    $\begingroup$ Array[a, {11, n, n}] . t^Range[-5,5], perhaps, assuming n is a positive Integer. There a bias in the index $i$: $(A_i)_{rc}$ is represented by a[i+6, r, c]. $\endgroup$
    – Michael E2
    Sep 14 at 14:00
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    $\begingroup$ small note: I think the Dot needs to be reversed in MichaelE2's answer; also note that you can eliminate the bias/offset by taking advantage of the second argument of Array, which gives the starting indices: (t^Range[-5, 5]) . Array[a, {11, n, n}, {-5, 1, 1}] $\endgroup$
    – thorimur
    Sep 15 at 1:37
  • $\begingroup$ @thorimur Thanks. Was answering from a phone and didn't check. $\endgroup$
    – Michael E2
    Sep 15 at 11:58
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For convenience, first define a function that creates the Ai with symbolic elements:

n = 2;
A[i_] := Array[(Subscript[a[i], #1, #2]) &, {n, n}]

The rest is trivial, simply sum it up:

Sum[A[i] t^i, {i, -5, 5}]

An example for n=2. Note that result is a matrix where every element is a Laurent series:

enter image description here

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