Performing Euler angle rotations in CAD systems (Q1344357)

The elementary result of this note can be extended and proved as follows. Let \(A\), \(B\) be the matrix representations of two linear transformations acting on a space \(V\) relative to a frame \(e\). Then the representation of the map of \(B\) in the frame \(Ae\) is \(B'= A^{-1} BA\). Hence, \(AB'= BA\). By induction, let \(A_ i'= (A_ 1\cdots A_{i-1})^{-1} A_ i(A_ 1\cdots A_{i- 1})\) be the representation of the map \(A_ i\) in the frame \(A_ 1\cdots A_{i- 1} e\). Then \(A_ 1 A_ 2'\cdots A_{i- 1}' A_ i'= A_ i\cdots A_ 1\). In particular, this is true for rotations given by Euler angles.