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In the case of $n = 2$, a hyperbolic rotation matrix by an arbitrary angle looks like:

$\begin{bmatrix} \cosh(\theta) & \sinh(\theta)\\ \sinh(\theta) & \cosh(\theta) \end{bmatrix}$

$\forall \theta \in \mathbb{R}^{1}$

These are Hermitian matrices with real entries. So is there a specific name/symbol for the n-dimensional real Hermitian matrix group? These are the hyperbolic equivalent of the SO(n) groups.

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I've seen the matrices you've written referred to as "hyperbolic rotations", and they/related matrices appear in discussions of the Lorentz group(s), like the Wikipedia entry for "rapidity".

In higher dimensions, I am not sure if the Lorentz group is what you're looking for, though.

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