I want to rotate a vector from the inertial J2000 frame into ITRF2008. I am using the NASA SPICE library, which provides a rotation matrix from J2000 to ITRF93 (they plan to upgrade the library to a modern ITRF in the near future but so far only ITRF93 is supported).
So my task is to then rotate from ITRF93 to ITRF2008. Given my vector $V_{J2000}$, I can easily compute $$ V_{ITRF93} = R_{ITRF93}^{J2000} V_{J2000} $$ where the rotation matrix $R_{ITRF93}^{J2000}$ is given by the NASA SPICE library. Now, according to http://itrf.ign.fr/doc_ITRF/Transfo-ITRF2008_ITRFs.txt, the transformation from ITRF93 to ITRF2008 is given as follows: $$ V_{ITRF93} = V_{ITRF2008} + T + M V_{ITRF2008} $$ where the vector $T$ and matrix $M$ are given in the link above. Rearranging this gives: $$ V_{ITRF2008} = (I + M)^{-1} (V_{ITRF93} - T) $$ so this amounts to a translation by $T$ followed by a matrix multiplication. However, one issue is that the matrix $I+M$ is not orthogonal (according to the parameters in the link above). So this is not really a rotation, but a transformation followed by a scaling.
It seems to me there should be an orthogonal rotation matrix allowing me to go from ITRF93 to ITRF2008: $$ V_{ITRF2008} = R_{ITRF2008}^{ITRF93} V_{ITRF93} $$ Does anyone know how to compute this rotation matrix?
My ultimate goal is to compute the spherical coordinate components of $V_{ITRF2008}$ (i.e. geocentric latitude, longitude and radius).
