Geometry of matrix differential systems (Q1071922)

The authors generalize some basic parts of the projective theory of linear homogeneous differential equations of order m,m\(\geq 2\), from the scalar to the matrix case. In Section 1, they define the (m-1)- dimensional left-projective space over the real \(n\times n\) matrices \(P=P_{m-1}(M_ n(R))\) and 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.