Matrices keep the original frame of reference. So when I rotate, then shear, the second shear would be in terms of the original frame and not the rotated frame.
However, in Euler angles, when I rotate in the x, then rotate in the z, the rotation in the z would be with respect to the new frame.
Here was my attempt to represent Euler Angles with rotation matrices:
$$R_x, R_y, R_z$$ are rotations around the fixed x-axis, y-axis, and z-axis.
First I rotate with around the x-axis: $$R_x$$ Then I rotate around the y-axis, but in the new frame:
$$(R_x R_y R_x^{-1}) R_x = R_xR_y$$
And finally around the z axis in the newest frame:
$$(R_xR_yR_zR_y^{-1}R_x^{-1})R_xR_y=R_xR_yR_z$$
Which is basically matrix rotation in reverse. Is there something wrong? It doesn't feel right.