If we have a real, symmetric, positive definite matrix $M$, then, by virtue of the Williamson decomposition, we can put it in the form: $$M=S^TDS$$ where $S$ is a symplectic matrix (ie. $SJS^T = J$ where $J$ is the symplectic form satisfying $J^T=-J$) and $D$ is a diagonal matrix. The diagonal values along $D$ are known as the symplectic eigenvalues of $M$.
For any real symmetric matrix, we can also write: $$M = R\Delta R^T$$ where $R$ is an orthogonal matrix and $\Delta$ is diagonal. The diagonal values along $\Delta$ are the regular eigenvalues of $M$.
My question is about the relationship between these two decompositions of $M$. In some cases, these decompositions are the same (ie. $M$ is diagonalised by a symplectic, orthogonal matrix and the symplectic eigenvalues are equal to the regular eigenvalues). In other cases, however, they are not.
I would like to know:
1) What are the conditions for the two decompositions to coincide? (so the symplectic eigenvalues equal the normal eigenvalues)
2) When they do not coincide, is there any relationship between the symplectic eigenvalues and the regular eigenvalues ?
Any help appreciated. Thanks in advance!