I was trying to understand how the transformation applied to a data set, translate to a covariance matrix, which seems fairly simple but I don't understand why I seem to have found two contradictory formulas.
On the robotics forum I found this which seemed to be a similar situation as the situation presented in this article (on page 82) but the answers are different :
For $x' = Rx$
On the forum the formula seem to be: $P_{x'} = RP_{x}R^t$
And in the article: $P_{x'} = JP_{x}J^t$ with $J$ the jacobian matrix.
So why are the formulas different, how are these different problems?
The two formulas are not inconsistent.
The formula involving the Jacobian is a general formula, accommodating (possibly) nonlinear transformations.
If the transformation is linear, as in your first formula, then the matrix R is the Jacobian of the transformation. So the first formula is a special case of the second formula.