不定双覆盖映射的反演
Inversion of the Indefinite Double Covering Map
摘要：
给出了不定双覆盖映射显式求逆的数学方法。这些都是基于给定矩阵的Givens分解或极分解在适当的，不确定的正交群与签名（p，q）。作为一个副产品，我们建立了覆盖群的原像，在适当的，不确定的正交群的正矩阵，本身可以被选为正定。反演相当于求解多个变量的多项式系统。我们的方法通过检查、Groebner基或通过反转相关的李代数同构并显式计算某些指数来解决这个系统。最后一种方法与Givens分解一起使用时，对所有（p，q）都是完全建设性的。其余的方法需要覆盖图的矩阵形式的细节，但随后提供关于原像的更多信息。该技术针对{（2，1），（2，2），（3，2），（4，1）}中的（p，q）进行说明。
英文摘要：
Algorithmic methods for the explicit inversion of the indefinite double covering maps are proposed. These are based on either the Givens decomposition or the polar decomposition of the given matrix in the proper, indefinite orthogonal group with signature (p,q). As a by-product we establish that the preimage in the covering group, of a positive matrix in the proper, indefinite orthogonal group, can itself be chosen to be positive definite. Inversion amounts to solving a polynomial system in several variables. Our methods solve this system by either inspection, Groebner bases or by inverting the associated Lie algebra isomorphism and computing certain exponentials explicitly. The last method extends fully constructively for all (p,q) when used together with Givens decompositions. The remaining methods require details of the matrix form of the covering map, but then provide more information about the preimage. The techniques are illustrated for (p, q)in {(2,1), (2,2), (3,2), (4,1)}.
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