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 , we associate a quadratic rational map to any such matrix A. The properties of the dynamical system obtained by iteration of , 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 , 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.