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I want to calculate an expression like $\left( M_1\otimes I+I\otimes M_2 \right) ^l$ with $M_i$'s symbolic matrices and $I$ the identity matrix with Mathematica. $M_i$'s are of the same dimension and my problem is how can I realize such an identity matrix without specifying the dimension of it, it should just work like a matrix multiply any matrix will still be the matrix itself(although it should be the same dimension as all the $M_i$)? Or is there some method that I can specify the dimension of the symbolic matrix $M_i$?

Thanks in advance!

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    $\begingroup$ After seeing the final edit, you can do something like this I guess. I forgot the exponent, but you get the general idea I think $\endgroup$
    – bmf
    Commented Feb 6, 2023 at 0:59
  • $\begingroup$ @bmf Thanks very much. I mean, is there some method that I can do not see the component of the matrix? For example, for $\left( M_1\otimes I+I\otimes M_2 \right) ^2$, the result I hope to see is $M_{1}^{2}\otimes I+I\otimes M_{2}^{2}+2M_1\otimes M_2$. $\endgroup$
    – narip
    Commented Feb 6, 2023 at 1:11
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    $\begingroup$ I guess you could try some global assumptions on the matrices and then do MatrixPower and TensorExpand or their cousins, but I have not managed to make it work. $\endgroup$
    – bmf
    Commented Feb 6, 2023 at 2:48

2 Answers 2

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I think you can use KroneckerProduct and TensorExpand. Define:

e = KroneckerProduct[M1, IdentityMatrix[d]] + KroneckerProduct[IdentityMatrix[d], M2];

Then:

TensorExpand[e . e, Assumptions -> (M1|M2) ∈ Matrices[{d, d}]]

2 KroneckerProduct[M1, M2] + KroneckerProduct[IdentityMatrix[d], MatrixPower[M2, 2]] + KroneckerProduct[MatrixPower[M1, 2], IdentityMatrix[d]]

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  • $\begingroup$ What about MatrixPower[e,l]? $\endgroup$
    – narip
    Commented Feb 7, 2023 at 0:22
  • $\begingroup$ @narip Can you be more specific about your question? Is it that you want to see M1 . M1 instead of MatrixPower[M1, 2]? $\endgroup$
    – Carl Woll
    Commented Feb 7, 2023 at 0:30
  • $\begingroup$ Oh, thanks. I mean, if I want to see e.e.e or more e.e.e.e, how can I do this? I try MatrixPower[e,3], but the result of Mathematica also contains MatrixPower[e,3]. $\endgroup$
    – narip
    Commented Feb 7, 2023 at 1:19
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Note that the symbol "[CircleTimes]" has not predefined meaning.Therefore, the terms are algebraically simply treated as one object. Further, "I" has built in meaning of Sqrt[-1], do not use it for a variable name. Therefore, by writing e.g.:

(M1⊗II + II⊗M2)^2 // Expand

you get:

enter image description here

However, to get your expected result, you assume some properties that you did not declare.

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