Geometries of the projective matrix space, II

AbstractWe have recently investigated the matrix projective line. Our interest was focused at the Möbius transformations W(Z) = (ZC + D)−1(ZA + B), where A, B, C, D, Z are n × n complex matrices. We were led to generalize the complex plane, the Riemann sphere and the unit disk. Our first task is to study the spherical, Euclidean, and non-Euclidean circles in terms of points at given distance from some point, and in terms of zeroes of some (Hermitian) quadratic form. The relations between the various circles are discussed. We also study Möbius transformations that carry the unit disk into itself, as well as those that carry (or interchange) Hermitian and unitary matrices. Finally, we introduce the stereographic projection from the generalized Riemann sphere to the Euclidean plane, and show that some spherical circles are mapped onto Euclidean ones