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I have a tensor

    T = Array[Subscript[K, ##] &, {2, 2, 2, 2}]

And its matrix form is $$ \left( \begin{array}{cc} \left( \begin{array}{cc} K_{1,1,1,1} & K_{1,1,1,2} \\ K_{1,1,2,1} & K_{1,1,2,2} \\ \end{array} \right) & \left( \begin{array}{cc} K_{1,2,1,1} & K_{1,2,1,2} \\ K_{1,2,2,1} & K_{1,2,2,2} \\ \end{array} \right) \\ \left( \begin{array}{cc} K_{2,1,1,1} & K_{2,1,1,2} \\ K_{2,1,2,1} & K_{2,1,2,2} \\ \end{array} \right) & \left( \begin{array}{cc} K_{2,2,1,1} & K_{2,2,1,2} \\ K_{2,2,2,1} & K_{2,2,2,2} \\ \end{array} \right) \\ \end{array} \right) $$

I use

    ArrayReshape[T, {4, 4}]

and get the matrix

$$ \left( \begin{array}{cccc} K_{1,1,1,1} & K_{1,1,1,2} & K_{1,1,2,1} & K_{1,1,2,2} \\ K_{1,2,1,1} & K_{1,2,1,2} & K_{1,2,2,1} & K_{1,2,2,2} \\ K_{2,1,1,1} & K_{2,1,1,2} & K_{2,1,2,1} & K_{2,1,2,2} \\ K_{2,2,1,1} & K_{2,2,1,2} & K_{2,2,2,1} & K_{2,2,2,2} \\ \end{array} \right). $$

However, I want the matrix looks exactly like the matrix form of the tensor T. How can I do it in Mathematica?

Ps: This can be done by

    ArrayFlatten[T]

but When the Dimension of T grows, it doesn't works. For example, if

    T = Array[Subscript[K, ##] &, {2, 2, 2, 2, 2, 2}]

then I need to use

    ArrayFlatten[ArrayFlatten[T]]

to get the same elements matrix as the matrixform of the tensor T.

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1 Answer 1

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$\begingroup$
T = Array[Subscript[K, ##] &, {2, 2, 2, 2}];
Flatten[T, {{1, 3}, {2, 4}}] // MatrixForm

$$ \left( \begin{array}{cccc} K_{1,1,1,1} & K_{1,1,1,2} & K_{1,2,1,1} & K_{1,2,1,2} \\ K_{1,1,2,1} & K_{1,1,2,2} & K_{1,2,2,1} & K_{1,2,2,2} \\ K_{2,1,1,1} & K_{2,1,1,2} & K_{2,2,1,1} & K_{2,2,1,2} \\ K_{2,1,2,1} & K_{2,1,2,2} & K_{2,2,2,1} & K_{2,2,2,2} \\ \end{array} \right) $$

T = Array[Subscript[K, ##] &, {2, 2, 2, 2, 2, 2}];
Flatten[T, {{1, 3, 5}, {2, 4, 6}}] // MatrixForm

$$ \left( \begin{array}{cccccccc} K_{1,1,1,1,1,1} & K_{1,1,1,1,1,2} & K_{1,1,1,2,1,1} & K_{1,1,1,2,1,2} & K_{1,2,1,1,1,1} & K_{1,2,1,1,1,2} & K_{1,2,1,2,1,1} & K_{1,2,1,2,1,2} \\ K_{1,1,1,1,2,1} & K_{1,1,1,1,2,2} & K_{1,1,1,2,2,1} & K_{1,1,1,2,2,2} & K_{1,2,1,1,2,1} & K_{1,2,1,1,2,2} & K_{1,2,1,2,2,1} & K_{1,2,1,2,2,2} \\ K_{1,1,2,1,1,1} & K_{1,1,2,1,1,2} & K_{1,1,2,2,1,1} & K_{1,1,2,2,1,2} & K_{1,2,2,1,1,1} & K_{1,2,2,1,1,2} & K_{1,2,2,2,1,1} & K_{1,2,2,2,1,2} \\ K_{1,1,2,1,2,1} & K_{1,1,2,1,2,2} & K_{1,1,2,2,2,1} & K_{1,1,2,2,2,2} & K_{1,2,2,1,2,1} & K_{1,2,2,1,2,2} & K_{1,2,2,2,2,1} & K_{1,2,2,2,2,2} \\ K_{2,1,1,1,1,1} & K_{2,1,1,1,1,2} & K_{2,1,1,2,1,1} & K_{2,1,1,2,1,2} & K_{2,2,1,1,1,1} & K_{2,2,1,1,1,2} & K_{2,2,1,2,1,1} & K_{2,2,1,2,1,2} \\ K_{2,1,1,1,2,1} & K_{2,1,1,1,2,2} & K_{2,1,1,2,2,1} & K_{2,1,1,2,2,2} & K_{2,2,1,1,2,1} & K_{2,2,1,1,2,2} & K_{2,2,1,2,2,1} & K_{2,2,1,2,2,2} \\ K_{2,1,2,1,1,1} & K_{2,1,2,1,1,2} & K_{2,1,2,2,1,1} & K_{2,1,2,2,1,2} & K_{2,2,2,1,1,1} & K_{2,2,2,1,1,2} & K_{2,2,2,2,1,1} & K_{2,2,2,2,1,2} \\ K_{2,1,2,1,2,1} & K_{2,1,2,1,2,2} & K_{2,1,2,2,2,1} & K_{2,1,2,2,2,2} & K_{2,2,2,1,2,1} & K_{2,2,2,1,2,2} & K_{2,2,2,2,2,1} & K_{2,2,2,2,2,2} \\ \end{array} \right) $$

Et cetera. For example, for a rank-8 tensor,

n = 8;
T = Array[Subscript[K, ##] &, ConstantArray[2, n]];
Flatten[T, {Range[1, n, 2], Range[2, n, 2]}] // MatrixForm

$$ \left( \begin{array}{cccccccccccccccc} K_{1,1,1,1,1,1,1,1} & K_{1,1,1,1,1,1,1,2} & K_{1,1,1,1,1,2,1,1} & K_{1,1,1,1,1,2,1,2} & K_{1,1,1,2,1,1,1,1} & K_{1,1,1,2,1,1,1,2} & K_{1,1,1,2,1,2,1,1} & K_{1,1,1,2,1,2,1,2} & K_{1,2,1,1,1,1,1,1} & K_{1,2,1,1,1,1,1,2} & K_{1,2,1,1,1,2,1,1} & K_{1,2,1,1,1,2,1,2} & K_{1,2,1,2,1,1,1,1} & K_{1,2,1,2,1,1,1,2} & K_{1,2,1,2,1,2,1,1} & K_{1,2,1,2,1,2,1,2} \\ K_{1,1,1,1,1,1,2,1} & K_{1,1,1,1,1,1,2,2} & K_{1,1,1,1,1,2,2,1} & K_{1,1,1,1,1,2,2,2} & K_{1,1,1,2,1,1,2,1} & K_{1,1,1,2,1,1,2,2} & K_{1,1,1,2,1,2,2,1} & K_{1,1,1,2,1,2,2,2} & K_{1,2,1,1,1,1,2,1} & K_{1,2,1,1,1,1,2,2} & K_{1,2,1,1,1,2,2,1} & K_{1,2,1,1,1,2,2,2} & K_{1,2,1,2,1,1,2,1} & K_{1,2,1,2,1,1,2,2} & K_{1,2,1,2,1,2,2,1} & K_{1,2,1,2,1,2,2,2} \\ K_{1,1,1,1,2,1,1,1} & K_{1,1,1,1,2,1,1,2} & K_{1,1,1,1,2,2,1,1} & K_{1,1,1,1,2,2,1,2} & K_{1,1,1,2,2,1,1,1} & K_{1,1,1,2,2,1,1,2} & K_{1,1,1,2,2,2,1,1} & K_{1,1,1,2,2,2,1,2} & K_{1,2,1,1,2,1,1,1} & K_{1,2,1,1,2,1,1,2} & K_{1,2,1,1,2,2,1,1} & K_{1,2,1,1,2,2,1,2} & K_{1,2,1,2,2,1,1,1} & K_{1,2,1,2,2,1,1,2} & K_{1,2,1,2,2,2,1,1} & K_{1,2,1,2,2,2,1,2} \\ K_{1,1,1,1,2,1,2,1} & K_{1,1,1,1,2,1,2,2} & K_{1,1,1,1,2,2,2,1} & K_{1,1,1,1,2,2,2,2} & K_{1,1,1,2,2,1,2,1} & K_{1,1,1,2,2,1,2,2} & K_{1,1,1,2,2,2,2,1} & K_{1,1,1,2,2,2,2,2} & K_{1,2,1,1,2,1,2,1} & K_{1,2,1,1,2,1,2,2} & K_{1,2,1,1,2,2,2,1} & K_{1,2,1,1,2,2,2,2} & K_{1,2,1,2,2,1,2,1} & K_{1,2,1,2,2,1,2,2} & K_{1,2,1,2,2,2,2,1} & K_{1,2,1,2,2,2,2,2} \\ K_{1,1,2,1,1,1,1,1} & K_{1,1,2,1,1,1,1,2} & K_{1,1,2,1,1,2,1,1} & K_{1,1,2,1,1,2,1,2} & K_{1,1,2,2,1,1,1,1} & K_{1,1,2,2,1,1,1,2} & K_{1,1,2,2,1,2,1,1} & K_{1,1,2,2,1,2,1,2} & K_{1,2,2,1,1,1,1,1} & K_{1,2,2,1,1,1,1,2} & K_{1,2,2,1,1,2,1,1} & K_{1,2,2,1,1,2,1,2} & K_{1,2,2,2,1,1,1,1} & K_{1,2,2,2,1,1,1,2} & K_{1,2,2,2,1,2,1,1} & K_{1,2,2,2,1,2,1,2} \\ K_{1,1,2,1,1,1,2,1} & K_{1,1,2,1,1,1,2,2} & K_{1,1,2,1,1,2,2,1} & K_{1,1,2,1,1,2,2,2} & K_{1,1,2,2,1,1,2,1} & K_{1,1,2,2,1,1,2,2} & K_{1,1,2,2,1,2,2,1} & K_{1,1,2,2,1,2,2,2} & K_{1,2,2,1,1,1,2,1} & K_{1,2,2,1,1,1,2,2} & K_{1,2,2,1,1,2,2,1} & K_{1,2,2,1,1,2,2,2} & K_{1,2,2,2,1,1,2,1} & K_{1,2,2,2,1,1,2,2} & K_{1,2,2,2,1,2,2,1} & K_{1,2,2,2,1,2,2,2} \\ K_{1,1,2,1,2,1,1,1} & K_{1,1,2,1,2,1,1,2} & K_{1,1,2,1,2,2,1,1} & K_{1,1,2,1,2,2,1,2} & K_{1,1,2,2,2,1,1,1} & K_{1,1,2,2,2,1,1,2} & K_{1,1,2,2,2,2,1,1} & K_{1,1,2,2,2,2,1,2} & K_{1,2,2,1,2,1,1,1} & K_{1,2,2,1,2,1,1,2} & K_{1,2,2,1,2,2,1,1} & K_{1,2,2,1,2,2,1,2} & K_{1,2,2,2,2,1,1,1} & K_{1,2,2,2,2,1,1,2} & K_{1,2,2,2,2,2,1,1} & K_{1,2,2,2,2,2,1,2} \\ K_{1,1,2,1,2,1,2,1} & K_{1,1,2,1,2,1,2,2} & K_{1,1,2,1,2,2,2,1} & K_{1,1,2,1,2,2,2,2} & K_{1,1,2,2,2,1,2,1} & K_{1,1,2,2,2,1,2,2} & K_{1,1,2,2,2,2,2,1} & K_{1,1,2,2,2,2,2,2} & K_{1,2,2,1,2,1,2,1} & K_{1,2,2,1,2,1,2,2} & K_{1,2,2,1,2,2,2,1} & K_{1,2,2,1,2,2,2,2} & K_{1,2,2,2,2,1,2,1} & K_{1,2,2,2,2,1,2,2} & K_{1,2,2,2,2,2,2,1} & K_{1,2,2,2,2,2,2,2} \\ K_{2,1,1,1,1,1,1,1} & K_{2,1,1,1,1,1,1,2} & K_{2,1,1,1,1,2,1,1} & K_{2,1,1,1,1,2,1,2} & K_{2,1,1,2,1,1,1,1} & K_{2,1,1,2,1,1,1,2} & K_{2,1,1,2,1,2,1,1} & K_{2,1,1,2,1,2,1,2} & K_{2,2,1,1,1,1,1,1} & K_{2,2,1,1,1,1,1,2} & K_{2,2,1,1,1,2,1,1} & K_{2,2,1,1,1,2,1,2} & K_{2,2,1,2,1,1,1,1} & K_{2,2,1,2,1,1,1,2} & K_{2,2,1,2,1,2,1,1} & K_{2,2,1,2,1,2,1,2} \\ K_{2,1,1,1,1,1,2,1} & K_{2,1,1,1,1,1,2,2} & K_{2,1,1,1,1,2,2,1} & K_{2,1,1,1,1,2,2,2} & K_{2,1,1,2,1,1,2,1} & K_{2,1,1,2,1,1,2,2} & K_{2,1,1,2,1,2,2,1} & K_{2,1,1,2,1,2,2,2} & K_{2,2,1,1,1,1,2,1} & K_{2,2,1,1,1,1,2,2} & K_{2,2,1,1,1,2,2,1} & K_{2,2,1,1,1,2,2,2} & K_{2,2,1,2,1,1,2,1} & K_{2,2,1,2,1,1,2,2} & K_{2,2,1,2,1,2,2,1} & K_{2,2,1,2,1,2,2,2} \\ K_{2,1,1,1,2,1,1,1} & K_{2,1,1,1,2,1,1,2} & K_{2,1,1,1,2,2,1,1} & K_{2,1,1,1,2,2,1,2} & K_{2,1,1,2,2,1,1,1} & K_{2,1,1,2,2,1,1,2} & K_{2,1,1,2,2,2,1,1} & K_{2,1,1,2,2,2,1,2} & K_{2,2,1,1,2,1,1,1} & K_{2,2,1,1,2,1,1,2} & K_{2,2,1,1,2,2,1,1} & K_{2,2,1,1,2,2,1,2} & K_{2,2,1,2,2,1,1,1} & K_{2,2,1,2,2,1,1,2} & K_{2,2,1,2,2,2,1,1} & K_{2,2,1,2,2,2,1,2} \\ K_{2,1,1,1,2,1,2,1} & K_{2,1,1,1,2,1,2,2} & K_{2,1,1,1,2,2,2,1} & K_{2,1,1,1,2,2,2,2} & K_{2,1,1,2,2,1,2,1} & K_{2,1,1,2,2,1,2,2} & K_{2,1,1,2,2,2,2,1} & K_{2,1,1,2,2,2,2,2} & K_{2,2,1,1,2,1,2,1} & K_{2,2,1,1,2,1,2,2} & K_{2,2,1,1,2,2,2,1} & K_{2,2,1,1,2,2,2,2} & K_{2,2,1,2,2,1,2,1} & K_{2,2,1,2,2,1,2,2} & K_{2,2,1,2,2,2,2,1} & K_{2,2,1,2,2,2,2,2} \\ K_{2,1,2,1,1,1,1,1} & K_{2,1,2,1,1,1,1,2} & K_{2,1,2,1,1,2,1,1} & K_{2,1,2,1,1,2,1,2} & K_{2,1,2,2,1,1,1,1} & K_{2,1,2,2,1,1,1,2} & K_{2,1,2,2,1,2,1,1} & K_{2,1,2,2,1,2,1,2} & K_{2,2,2,1,1,1,1,1} & K_{2,2,2,1,1,1,1,2} & K_{2,2,2,1,1,2,1,1} & K_{2,2,2,1,1,2,1,2} & K_{2,2,2,2,1,1,1,1} & K_{2,2,2,2,1,1,1,2} & K_{2,2,2,2,1,2,1,1} & K_{2,2,2,2,1,2,1,2} \\ K_{2,1,2,1,1,1,2,1} & K_{2,1,2,1,1,1,2,2} & K_{2,1,2,1,1,2,2,1} & K_{2,1,2,1,1,2,2,2} & K_{2,1,2,2,1,1,2,1} & K_{2,1,2,2,1,1,2,2} & K_{2,1,2,2,1,2,2,1} & K_{2,1,2,2,1,2,2,2} & K_{2,2,2,1,1,1,2,1} & K_{2,2,2,1,1,1,2,2} & K_{2,2,2,1,1,2,2,1} & K_{2,2,2,1,1,2,2,2} & K_{2,2,2,2,1,1,2,1} & K_{2,2,2,2,1,1,2,2} & K_{2,2,2,2,1,2,2,1} & K_{2,2,2,2,1,2,2,2} \\ K_{2,1,2,1,2,1,1,1} & K_{2,1,2,1,2,1,1,2} & K_{2,1,2,1,2,2,1,1} & K_{2,1,2,1,2,2,1,2} & K_{2,1,2,2,2,1,1,1} & K_{2,1,2,2,2,1,1,2} & K_{2,1,2,2,2,2,1,1} & K_{2,1,2,2,2,2,1,2} & K_{2,2,2,1,2,1,1,1} & K_{2,2,2,1,2,1,1,2} & K_{2,2,2,1,2,2,1,1} & K_{2,2,2,1,2,2,1,2} & K_{2,2,2,2,2,1,1,1} & K_{2,2,2,2,2,1,1,2} & K_{2,2,2,2,2,2,1,1} & K_{2,2,2,2,2,2,1,2} \\ K_{2,1,2,1,2,1,2,1} & K_{2,1,2,1,2,1,2,2} & K_{2,1,2,1,2,2,2,1} & K_{2,1,2,1,2,2,2,2} & K_{2,1,2,2,2,1,2,1} & K_{2,1,2,2,2,1,2,2} & K_{2,1,2,2,2,2,2,1} & K_{2,1,2,2,2,2,2,2} & K_{2,2,2,1,2,1,2,1} & K_{2,2,2,1,2,1,2,2} & K_{2,2,2,1,2,2,2,1} & K_{2,2,2,1,2,2,2,2} & K_{2,2,2,2,2,1,2,1} & K_{2,2,2,2,2,1,2,2} & K_{2,2,2,2,2,2,2,1} & K_{2,2,2,2,2,2,2,2} \\ \end{array} \right) $$

$\endgroup$
4
  • $\begingroup$ How to do the inverse operation? From the Matrix to the same arrange tensor? $\endgroup$
    – LX.CC
    Aug 31, 2022 at 7:36
  • $\begingroup$ The inverse can be done by ``` Partition[Flatten[T, {{1, 3}, {2, 4}}] , {2, 2}]```! $\endgroup$
    – LX.CC
    Aug 31, 2022 at 7:47
  • $\begingroup$ But for the high dimension, I need to do Partition[Partition[...]]. Is there any way to do this? $\endgroup$
    – LX.CC
    Aug 31, 2022 at 7:52
  • 2
    $\begingroup$ Please ask a separate question. $\endgroup$
    – Roman
    Aug 31, 2022 at 8:01

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