Illustrative Study on Linear Differential Equations and Matrix Polynomials

The objective of this paper is to study the theoretical analysis of linear differential-algebraic equations (DAEs) of higher order as well as the regularity and singularity of matrix polynomials. Some invariants and condensed forms under appropriate equivalent transformations are established for systems of linear higher-order DAEs with constant and variable coefficients. Inductively, based on condensed forms, the properties of corresponding system of DAEs are studied and by differentiation-and-elimination steps reduce the system to a simpler but equivalent system. It includes the consistency of initial conditions and unique solvability of the original DAE system. It is shown that the following equivalence holds for a DAE system with strangeness-index μ and square and constant coefficients: for any consistent initial condition and any right-hand side f(t)∈ Cμ([t0, t1],Cn) the associated initial value problem has a unique solution if and only if the matrix polynomial associated with the system is regular. Moreover, some necessary and sufficient conditions for column-and-row regularity and singularity of rectangular matrix polynomials are derived. A geometrical characterization of singular matrix pencils is also described. Furthermore, an algorithm is presented which using rank information about the coefficient matrices and via computing determinants decides whether a given matrix polynomial is regular. This paper basically focuses on the connection between the solution behaviour of a system of DAEs and regularity or singularity of the matrix polynomial associated with that system.