The Orthogonal Matrices of O(2) under A Transitive Standard Action of S^1

In this paper, we study a Lie group action of the matrix Lie group O(2) on S1 the unit sphere . The research aims to establish the explicit formulas for all entries of whose action on S1 is transitive. All possibilities matrices of are given in which the space is homogeneous. We prove that there are exactly two matrices in such that is the homogeneous space. Moreover, the homogeneous spaces S(n-1) of O(n) for n=3 are also discussed.