When carrying out a rotation of the co-ordinate axes, x1 x2 x3 to say x1' x2' x3',
When dealinf with the rotation of axes about a vector (using an orthogonal rotation matrix R), I was taught to use V' = R.V BUT for a matrix I should use M' = R.M.R^T (where R^-1 = R^T)
However, if my matrix is say M =
$$ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} $$
I.e. a crystal with unit vectors: V1 $$ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} $$ V2 $$ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} $$
V3 $$ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} $$ enter image description here Then this obviously does not work as it just takes me back tot he original identity rather than rotating my matrix. So, is the identity matrix an exception here i.e. should I use M'=R.M only when my matrix is the identity and use M'=R.M.R^T when it is some other matrix e.g. if my crystal took different unit vecors like M = $$ \begin{matrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 4 \\ \end{matrix} $$ This is really confusing me.
