Square Roots of Symplectic Matrices

2026/07/21 by Qinglong Zhou · 1 voice
#math.SG

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Abstract

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.

Discussions

• Qinglong Zhou: Square Roots of Symplectic Matrices https://arxiv.org/abs/2607.19094 https://arxiv.org/pdf/2607.19094 https://arxiv.org/html/2607.19094 [bsky, 0 points, 0 comments]

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