$\Omega_1= \begin{pmatrix} 1 & 0.8 & -0.8\\ 0.8 & 1 & \rho_1\\ -0.8 & \rho_1 & 1 \end{pmatrix}$ and $\Omega_2= \begin{pmatrix} 1 & 0.4 & -0.4\\ 0.4 & 1 & \rho_2\\ -0.4 & \rho_2 & 1 \end{pmatrix} $ are 2 correlation matrices.
Using Sylvester's criterion, I got: $-1 \leq \rho_1 \leq -0.28$ and $-1 \leq \rho_2 \leq 0.68$
However, when I look at these correlation matrices, I feel that they have sort of relationship, hence, same for $\rho_1$ and $\rho_2$.
Is it wrong to say: $\rho_1=2\,\rho_2?$
Any intuition about it? Thanks
Yes, it is wrong. $\Omega_1$ and $\Omega_2$ are correlation matrixes for any $\rho_1$ and $\rho_2$ from the intervals that you wrote. If we are not informed what triples of r.v. have this correlation matrixes and how this triples are related, one cannot provide further conclusions.