In a paper I am reading, I found a result that I have been trying to work through but I am a bit stuck. The paper states that the expression for a transformation is:
$ f = M^{-1}(S*x - T)$
where f is the coordinates of a point in one coordinate system in R3, x is the coordinates in a different coordinate system in R3, S is a scaling matrix, T is a translation vector, and M is a rotation matrix. Then the author goes ahead and states that because the transpose of a rotation matrix is equal to its inverse, the following expression can be derived:
$ ||T - S*x||^2 = ||f||^2 $
I am not exactly sure how this expression was derived. I was hoping someone could provide some insight. I tried working through it using a dot product definition for the norm indicated, but I am not sure how that would help.
This is coming from
$$\begin{aligned} \Vert f \Vert^2 &= f^Tf = [M^{-1}(S*x - T)]^T [M^{-1}(S*x - T)]\\ &=(S*x - T)^T (M^{-1})^TM^{-1}(S*x - T)\\ &=(S*x - T)^T MM^{-1}(S*x - T)\\ &=(S*x - T)^T (S*x - T) = \Vert S*x-T \Vert^2 \end{aligned}$$