Suppose we have two coordinate systems with different bases. Conversion of translations can be trivial, but the conversion of rotation is less. One can resort to quaternion multiplication, but that requires additional packages and understandings. Here we give a simple derivation using matrices.

Say we know the transformation between bases from system 1 to 2: $T_1^2$. For instance, we know a transformation from the CV system to a CG system by

$$ T_{CV}^{CG} = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} $$

where the y-axis is flipped.

Suppose we have a homogeneous point $x_1=[x,y,z,1]^\top$ in system 1, then its correspondence in system 2 is

$$ x_2 = T_1^2 x_1. $$

Now suppose we have a rigid transformation $T_1$ (or pose) in system 1 consisting of a rotation and a translation. The transformed $x_1$ is then

$$ x_1' = T_1 x_1, $$

and its correspondence in system 2 is

$$ x_2' = T_1^2 x_1' = T_1^2T_1x_1. $$

On the other hand, we can directly transform it using $T_2$ (our target, the transformed transformation that includes rotation and translation) natively in system 2:

$$ x_2' = T_2 x_2 = T_2 T_1^2 x_1. $$

Therefore we know for arbitrary $x_1$ in system 1,

$$ T_1^2T_1x_1 = T_2T_1^2x_1,\\ T_2 = T_1^2 T_1 {T_1^2}^{-1}. $$