Simultaneous symplectic spectral decomposition of positive semidefinite matrices

We establish necessary and sufficient conditions on simultaneous symplectic spectral decomposition of a family of $2n\times 2n$ real positive semidefinite matrices with symplectic kernels. We also provide a precise algebraic condition on a $2n\times 2n$ real positive semidefinite matrix with symplectic kernel for orthosymplectic spectral diagonalization, which generalizes a known result for positive definite matrices.