Let O2 be the Lie grouo consists of all 2 by 2 orthogonal matrices, i.e all matrices such that their transpose is equal to their inverses. The operation is the usual product of matrices. It falls into two pieces; The matrices with detreminant 1 which forms a subgroup of O2 and the matrices with detreminant -1. We can interpret the first case as a rotation by theta (where theta = 0 is the identity), and the second as a reflection of the identinty across a line at an angle of theta/2. So, we can express any orthogonal matrix in terms of theta where theta is between 0 and 2pi. My question is if we have two reflection matrices A_1 and A_2, how to express the matrix (A_1 times A_2) in terms of theta? Ofcourse (A_1 times A_2) is a rotation matrix. Samething how to get theta for the matrix which is the product of two rotation matrices; the product of a rotation matix with a reflection matrix.
Thank you in advance