线性映射何时至少属于一个正交群或辛群？
When does a linear map belong to at least one orthogonal or symplectic group?
摘要：
给定任意域K上有限维向量空间的自同态u，我们给出了正则二次型存在的充要条件（分别为.辛形式），其中u是正交的（分别为辛）。这个问题的解决方案在char（K）<>2的情况下是已知的，我们的主要贡献在于char（K）=2的情况。当char（K）=2时，给出了u正交的正则对称双线性型存在的充要条件.在K是特征为2的有限情形下，我们给出了双曲二次型存在的充要条件（分别为正则非双曲型二次型，正则非交替对称双线性型），其中u是正交的。
英文摘要：
Given an endomorphism u of a finite-dimensional vector space over an arbitrary field K, we give necessary and sufficient conditions for the existence of a regular quadratic form (resp. a symplectic form) for which u is orthogonal (resp. symplectic). A solution to this problem being already known in the case char(K)<>2, our main contribution lies in the case char(K)=2. When char(K)=2, we also give necessary and sufficient conditions for the existence of a regular symmetric bilinear form for which u is orthogonal. In the case K is finite with characteristic 2, we give necessary and sufficient conditions for the existence of an hyperbolic quadratic form (resp. a regular non-hyperbolic quadratic form, resp. a regular non-alternate symmetric bilinear form) for which u is orthogonal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
