how to get the angle theta of a product of two orthogonal matrices

Hi;

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

1 Attachment(s)

Re: how to get the angle theta of a product of two orthogonal matrices

Your product is just a rotation matrix of an angle that is the sum of the individual rotations.

Attachment 30022

Re: how to get the angle theta of a product of two orthogonal matrices

I know that a rotation by an angle theta is given by multiplication of e^(i.theta). so the product of two rotation matrix gives us the rotation matrix with theta just the sum of the two thetas. Thanks for help.

Now how to get theta for the product of rotation matrix and reflection matrix or for tow reflection matrices. I don't think that it is the sum of the two thetas.

1 Attachment(s)

Re: how to get the angle theta of a product of two orthogonal matrices

Hi,

I think the attachment shows what you want.

Attachment 30024

Re: how to get the angle theta of a product of two orthogonal matrices

Thank you so much johng for enlightening me and sorry for late reply.