Square Roots of Symplectic Matrices

This paper characterizes the existence of real symplectic square roots for symplectic matrices. The decomposition of Wonenburger matrices with respect to their eigenvalues permits a partition of the problem into three primary cases. A symplectic matrix whose spectrum avoids the negative real axis is shown to always admit a real symplectic square root. For a negative hyperbolic matrix, such a root exists if and only if the half-dimension is even and the matrix itself is a $\diamond$-square. In the degenerate case of eigenvalue $-1$, a necessary and sufficient condition is established via a decomposition into specific standard blocks.