Geometry of matrix differential systems

Abstract

In this paper we generalize some basic parts of the projective theory of linear homogeneous differential equations of order m, m > 2, from the scalar to the matrix case. In Section 1 we define the (m l)-dimensional left-projective space over the real n x n matrices P = P, ~ [(M,(R)) and, relying on a previous paper [7] and on standard methods of linear algebra, we state the basic properties of this space. In Section 2 the geometry of the matrix differential systems is described. To each basis of matrix solutions of a given system there corresponds a curve in P and a change of basis corresponds to a projective mapping of this curve. A given curve, together with its projective maps, corresponds to a class of differential systems having projectively equivalent solutions. Some simple properties of these curves are established and a new kind of disconjugacy is defined for the matrix differential systems.