I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a hyperboloid with equation given by ## z^2 - x^2 - y^2 = 1##, while the rotation matrix is parameterized by a point on a circle. I guess the rotation matrix being parameterized by a point on a circle kind of makes sense intuitively, but I can't see how the hyperboloid can possibly parameterize the symmetric matrix. What am I missing here? If anyone can demonstrate the parametrization for the rotation matrix in a rigorous way aswell, I'd be glad.