Schwarz, Binyamin; Zaks, Abraham Geometries of the projective matrix space. II. (English) Zbl 0599.51008 J. Algebra 103, 686-707 (1986). Recently [cf. part I, ibid. 95, 263-307 (1985; Zbl 0589.51011)] the authors investigated the Euclidean, spherical and non-Euclidean metrics on suitable subspaces of the complex matrix projective line \(P_ 1(M_ n({\mathbb{C}}))\). Their interest is now focused at the Möbius transformations \(W(Z)=(ZC+D)^{-1}(ZA+B)\), where A, B, C, D, Z are \(n\times n\) complex matrices. The concepts ”complex plane”, ”Riemann sphere” and ”unit disk” are generalized accordingly. In chapter 1 the spherical, Euclidean and non-Euclidean circles are defined and studied in terms of (i) points at given distance from some point resp. (ii) zeroes of some (Hermitian) quadratic form. Chapter 2 deals with Möbius transformations that carry the unit disk into itself, as well as those that carry (or interchange) Hermitian and unitary matrices. Finally (chapter 3), the stereographic projection from the generalized Riemann sphere to the Euclidean plane is introduced; various properties of this map are dicussed. Reviewer: R.Koch Cited in 1 ReviewCited in 3 Documents MSC: 51B10 Möbius geometries 51N25 Analytic geometry with other transformation groups 20H20 Other matrix groups over fields 51M05 Euclidean geometries (general) and generalizations 51M10 Hyperbolic and elliptic geometries (general) and generalizations Keywords:Hermitian quadratic form; complex matrix projective line; spherical, Euclidean and non-Euclidean circles; Möbius transformations; stereographic projection; Riemann sphere Citations:Zbl 0589.51011 PDFBibTeX XMLCite \textit{B. Schwarz} and \textit{A. Zaks}, J. Algebra 103, 686--707 (1986; Zbl 0599.51008) Full Text: DOI References: [1] Carathéodory, C., (Theory of Functions of a Complex Variable, Vol. I (1954), Chelsea: Chelsea New York) [2] Scwharz, B.; Zaks, A., Matrix Möbius Transformations, Comm. Algebra, 9, 1913-1968 (1981) · Zbl 0479.51016 [3] Schwarz, B.; Zaks, A., Geometries of the projective matrix space, J. Algebra, 95, 263-307 (1985) · Zbl 0589.51011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.