Let's consider the action of the group $SL(n)$ in the projective space $\mathbb{P}(Sym(n))$ given by $g A= g A g^{T}$. Here $SL(n)$ is the special linear group and $Sym(n)$ the vector space of symmetric $n\times n$ matrices.
I would like to prove that invertible matrices belong in only one orbit, equivalently that given two regular symetric matrices $A, B$ there exists a matrix $g\in SL(n)$ such that $gA= \lambda B$ for some scalar $\lambda$.
Any hints or references in the direction of the proof I'm looking for?