If I have an arrow rotating around the y axis in it's own local space with matrix T, is it possible keep that rotation in world space consistent, but change the local matrix to a new matrix H so that this new matrix has the arrow rotating for example around the z axis? So in this example the change between the local spaces would be 90 degrees in x or:
$\begin{bmatrix}1 & 0 & 0\\0 & 0 & 1\\0&-1&0\end{bmatrix}$
Which if I understand I can get by T * inverse(H) for the difference between the rotation matrices but that's as far as I have gotten.