Let $\Sigma$ be a symmetric positive definite matrix with ones on the diagonal (= correlation matrix).
Let $A$ be an invertible matrix.
I'm pretty sure that if $\Omega:=A\Sigma A^t$ has ones on its diagonal, then
$$\Omega=Id\ \ \text{or}\ \ \Omega=\Sigma$$
(which would correspond to $A=Id$ or $A=chol(\Sigma)$)
But I don't know how to proove it. What do you think of this statement and could you help me with the proof ?
In other word, I think that the transformation $A\Sigma A^t$ of a correlation matrix $\Sigma$ can't give another non trivial correlation matrix.