From hyperbolic geometry to 2 × 2 Hermitian matrices and back

In this paper we show how the space SPD of 2 × 2 positive definite Hermitian matrices of determinant 1 can serve as a model for spatial hyperbolic geometry by defining an equivalence with the three-dimensional hyperboloid model embedded in four-dimensional Minkowski space. The new model provides new computational possibilities for hyperbolic geometry and conversely geometric tools are provided for the matrix theory. An illustration of the latter is carried out by transferring notions of center of mass and centroid to the matrix setting and developing basic properties they exhibit in that context.