Solution Properties of Linear Descriptor (Singular) Matrix Differential Systems of Higher Order with (Non-) Consistent Initial Conditions

and Applied Analysis 3 Definition 2.1. Given F,G ∈ Mnm and an indeterminate s ∈ F, the matrix pencil sF−G is called regular when m n and det sF −G / 0 where 0 is the zero element of M 1,F . In any other case, the pencil is called singular. In this paper, as we are going to see in the next paragraph, we consider the case that the pencil is singular. The next definition is very important, since the notion of strict equivalence between two pencils is presented. Definition 2.2. The pencil sF −G is said to be strictly equivalent to the pencil s ̃ F − ̃ G if and only if there exist nonsingular P ∈ Mn and Q ∈ Mm such that P sF −G Q s ̃ F − ̃ G. 2.1 The characterization of singular pencils requires the definition of additional sets of invariants known as the minimal indices. Let us assume that r rankF s sF − G , where F s denotes the field of rational functions in s having coefficients in the field F. The equations sF −G x s 0, ψ s sF −G 0 2.2 have nonzero solutions x s and ψ s which are vectors in the rational vector spaces Nright s Nright sF −G , Nleft s Nleft sF −G , 2.3 respectively, where Nright s { x s ∈ F s m : sF −G x s 0n } , Nleft s { ψ s ∈ F s n : ψ s sF −G 0Tm } . 2.4 The sets of the minimal degrees {vi, 1 ≤ i ≤ m−r} and {uj, 1 ≤ j ≤ n−r} are known as column minimal indices c.m.i. and row minimal indices r.m.i. of sF −G, respectively. Furthermore, if r rankF s sF −G , it is evident such that r m−r ∑