AI 中文总结
研究辛矩阵实辛平方根的存在性，通过Wonenburger矩阵按特征值分解分三种情形讨论，得出谱避开负实轴的辛矩阵总有实辛平方根等结论，建立了不同情形下实辛平方根存在的条件。
AI 中文摘要
本文刻画了辛矩阵实辛平方根的存在性。Wonenburger矩阵按特征值分解将问题分为三种主要情形。谱避开负实轴的辛矩阵总存在实辛平方根；对于负双曲矩阵，当且仅当半维数为偶数且矩阵本身是◇-平方时存在这样的根；在特征值为 -1 的退化情形下，通过分解为特定标准块建立了充要条件。
英文摘要
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.
Comments38 pages
