Matrices of linear forms and curves

In order to understand the triple action of PGLn+1 on the projective space of nonzero (n + 1) × (n + 1) matrices of linear forms on Pn, we associate a quadratic rational map ϕ:Pn→Pn to any such matrix A. The properties of the dynamical system obtained by iteration of ϕ, some of which are of a geometric nature, generate invariants and a canonical form for the orbit of A. We study a family of matrices parametrized by P1, whose associated geometry is given by the rational normal curve for each dimension n = 2, 3, 4. Our analysis involves the osculating flags to the curves; and we calculate the stabilizers of our rational maps and matrices.