Parametrization manifold of SL(2,R)

[Wledig]

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.

 

[fresh_42]

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.

Can you show the decomposition? I know of the Iwasawa decomposition into a product diagonal times upper triangle times rotation.

The rotation part is easy: A rotation matrix is of the form $K = \begin{bmatrix}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi\end{bmatrix}$ which is parameterized by $\varphi \in [0,2\pi )$.

The Iwasawa decomposition is $G=ANK$ with $A=\begin{bmatrix}e^t&0\\ 0 & e^{-t}\end{bmatrix}\; , \;N=\begin{bmatrix}1&x\\0&1\end{bmatrix}\,.$ Of course I could consider $AN$ but this isn't symmetric.

 

[Wledig]

Thanks for your reply. I think I should've mentioned this but the book also states that both matrices are unimodular. So we could have
$A=\begin{bmatrix}a&c\\ c & b\end{bmatrix}\$ and $K = \begin{bmatrix}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi\end{bmatrix}$ so that the product AK yields a matrix with determinant equal to that of the symmetric matrix which is 1 by the unimodular condition. The big question is how to relate the symmetric matrix to the hyperboloid though.

 

[fresh_42]

So the question boils down to the question what the coordinate transformation between
http://www.wolframalpha.com/input/?i=z^2-x^2-y^2=1
and
http://www.wolframalpha.com/input/?i=xy-z^2=1
is.

 

[Wledig]

I see, book would've done me a favor writing the equation in this second manner. Still, I guess I should've noticed that it was just a silly transformation. Thanks again for clarifying things.