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Given two matrices $A$ and $B$:

enter image description here

What transformation needs to be applied to transform matrix $A$ into matrix $B$?

A = {{0, c, b, -c + b c, a c, -a + a b, c + b c, a c, a + a b}, {0, c,
    b, c + b c, a c, a + a b, -c + b c, a c, -a + a b}, {0, -1, 0, b, 
   a, 0, b, a, 0}}

B = {{0, c, b, 0, c, b, 0, -1, 0}, {-c + b c, a c, -a + a b, c + b c, 
    a c, a + a b, b, a, 0}, {c + b c, a c, a + a b, -c + b c, 
    a c, -a + a b, b, a, 0}};

EDIT:

I found one method that only uses $\frac{dQ}{d\boldsymbol{\theta}}$, and the result is the same as direct differentiation $\frac{dQ^T}{d\boldsymbol{\theta}}$. The disadvantage of this method is the need to glue the matrix again. Maybe I can get around this with some kind of unified tensor operation ? To immediately receive the entire matrix, without additional gluing.

Rx = RotationMatrix[\[Phi][t], {1, 0, 0}];

Ry = RotationMatrix[\[Xi][t], {0, 1, 0}];

Rz = RotationMatrix[\[Psi][t], {0, 0, 1}];

Q = Rz.Ry.Rx;

v = {\[Phi][t], \[Xi][t], \[Psi][t]};

T1 = Flatten /@ D[Q, {v}];

T2 = Flatten /@ D[Transpose[Q], {v}];

P1 = {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0,
     0, 0, 0, 0, 1, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 
    1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 0, 0,
     0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}};

A1 = T1.{{1, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 1, 0}, {0, 0, 0}, {0, 0,
      0}, {0, 0, 1}, {0, 0, 0}, {0, 0, 0}};

A2 = T1.{{0, 0, 0}, {1, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 1, 0}, {0, 0,
      0}, {0, 0, 0}, {0, 0, 1}, {0, 0, 0}};

A3 = T1.{{0, 0, 0}, {0, 0, 0}, {1, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 1,
      0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 1}};

Transpose[P1.ArrayFlatten[{{A1}, {A2}, {A3}}]] == 
   Flatten /@ D[Transpose[Q], {v}] // MatrixForm;
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  • 1
    $\begingroup$ LinearSolve[A, B]? $\endgroup$
    – Lukas Lang
    Jul 23, 2021 at 11:03
  • $\begingroup$ @LukasLang please, form your comment as answer $\endgroup$
    – dtn
    Jul 23, 2021 at 11:05
  • $\begingroup$ @LukasLang looking at the matrices I have an assumption about permutation matrices, am I right? if so, what will they look like? $\endgroup$
    – dtn
    Jul 23, 2021 at 11:06

3 Answers 3

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B == ArrayReshape[Transpose[ArrayReshape[A, {3, 3, 3}]], {3, 9}]

True

Also, in loop style:

result = ConstantArray[0, {3, 9}];
Do[
  result[[i, 3 (j - 1) + k]] = A[[j, 3 (i - 1) + k]], 
  {i, 1, 3}, 
  {j, 1, 3}, 
  {k, 1, 3}
];
result == B

True

Or with a permutation matrix of size $27 \times 27$:

P = Block[{A, B, a},
   A = Array[a, {3, 9}];
   B = ArrayReshape[Transpose[ArrayReshape[A, {3, 3, 3}]], {3, 9}];
   SparseArray[D[Flatten[B], {Flatten[A], 1}]]
   ];

ArrayReshape[P.Flatten[A], Dimensions[B]]
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  • $\begingroup$ Thank you! is it possible to formalize your proposed operation? Express it as a formula? As I assumed, you are using a permutation. $\endgroup$
    – dtn
    Jul 23, 2021 at 12:17
  • 1
    $\begingroup$ Well, if you treat both matrices as 3-tensors ArrayReshape[A, {3, 3, 3}] and ArrayReshape[B, {3, 3, 3}], then it is just about permuting the first two slots of the tensors. $\endgroup$
    – Henrik Schumacher
    Jul 23, 2021 at 12:18
  • $\begingroup$ I mean the following. Find such a matrix $X$, or a pair of matrix $x$ and $X$ for example (I don't know exactly how many are needed), so that either $AX=B$ or $xAX=B$. $\endgroup$
    – dtn
    Jul 23, 2021 at 12:26
  • 1
    $\begingroup$ Hm. You certainly cannot do that if you require that both x and X are permutation matrices; this can be seen when you compare the two columns in A and B that contain the -1. $\endgroup$
    – Henrik Schumacher
    Jul 23, 2021 at 12:37
  • 1
    $\begingroup$ Not sure what you need this for. Is it about coding? Just in case that you mean to code that in C, C++ or whatever: note that dense matrices and arrays are stored in flattened form. So the permutation matrix P from my post, applied to the raw buffer of A would do what you want. $\endgroup$
    – Henrik Schumacher
    Jul 23, 2021 at 12:49
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You can use LinearSolve to find one possible matrix such that A.X==B:

X = LinearSolve[A, B];
A.X == B // FullSimplify
(* True *)
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  • $\begingroup$ The result is a matrix $X$? $\endgroup$
    – dtn
    Jul 23, 2021 at 11:08
  • $\begingroup$ @dtn Yes, X is a 9x9 matrix. If you are looking for another kind of "transformation", you'll have to be a bit more specific with your requirements. (When talking about transformations in the context of matrixes, the usual thing that's meant is a transformation matrix such as the X from the answer) $\endgroup$
    – Lukas Lang
    Jul 23, 2021 at 11:17
  • $\begingroup$ Thank you for your answer. I understand what you are talking about. At the moment, for my tasks there is only one requirement that transformations lie in the class of vector-matrix operations. But I will happily explore possible alternatives if you point me to the sources. $\endgroup$
    – dtn
    Jul 23, 2021 at 11:27
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    $\begingroup$ I don't think this can be done purely using permutation matrices and dot products. You need to allow some other operations for this to be possible $\endgroup$
    – Lukas Lang
    Jul 23, 2021 at 13:04
  • 1
    $\begingroup$ @dtn As far as I can tell, no such permutation matrices exist (assuming a permutation matrix is a matrix where each row/column has exactly one 1 and the rest is 0). A matrix product with such a matrix can only ever permute rows/columns, but that is not sufficient to perform the permutation you are after. $\endgroup$
    – Lukas Lang
    Jul 23, 2021 at 19:27
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Here is a solution using Part

Flatten[A[[All, # ;; ;; 3]] & /@ Range@3 // Transpose, 1] // Transpose
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  • $\begingroup$ math.stackexchange.com/a/4201431/656085 Please see the very end of the answer. I need something similar, but I don't know the structure of the $P_{9\times9}$ and $P_{3\times3}$ matrices. $\endgroup$
    – dtn
    Jul 23, 2021 at 15:32
  • $\begingroup$ See my edit please. I found a variant of the way how to switch to the transposed matrix through the original one. $\endgroup$
    – dtn
    Jul 23, 2021 at 19:45

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