Given two matrices $A$ and $B$:
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;
LinearSolve[A, B]
? $\endgroup$