Ciro Santilli $$Sponsor Ciro$$ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
🔗

# All indefinite orthogonal groups of matrices of equal metric signature are isomorphic

🔗
Following the definition of the indefinite orthogonal group, we want to show that only the metric signature matters.
🔗
First we can observe that the exact matrices are different. For example, taking the standard matrix of : $$[1001​] (218)$$ and: $$[2001​] (219)$$ both have the same metric signature. However, we notice that a rotation of 90 degrees, which preserves the first form, does not preserve the second one! E.g. consider the vector , then . But after a rotation of 90 degrees, it becomes , and now ! Therefore, we have to search for an isomorphism between the two sets of matrices.
🔗
For example, consider the orthogonal group, which can be defined as shown at the orthogonal group is the group of all matrices that preserve the dot product can be defined as:
🔗
🔗

🔗