It's probably dumb question, but I haven't used matrices for very long time and I really need that one.
I have two $3\times 3$ matrices $P$ and $C$, which have positive determinants and unknown transformation matrix T.
$$ T^TP^TPT = C^TC $$
Is there a way to compute $T$?
In general, the answer is no, since there can be many such $T$ that satisfy that equation. For instance, if $P = C = I$, then $T$ can be any orthogonal matrix, so there's no way to know which one without more information.
If you need to find all such $T$, multiply by $C^{-T}$ on the left and $C^T$ on the right. The equation becomes $$ C^{-T}T^TP^T PTC^{-1} = (PTC^{-1})^T(PTC^{-1}) = I $$
In this case take $Q$ to be any orthogonal matrix, and simply take $$ PTC^{-1} = Q$$ so that we have $$ T = P^{-1}QC $$
If you just need one such, $T$, the simplest case is that $Q = I$, whence $$ T = P^{-1}C $$ gives a solution.
