The index and split forms of linear differential-algebraic equations

We consider linear systems of ordinary differential equations (ODE) with rectangular matrices of coefficients, including the case when the matrix before the derivative of the desired vector function is not full rank for all argument values from the domain. Systems of this type are usually called differential-algebraic equations (DAEs). We obtained criteria for the existence of nonsingular transformations splitting the system into subsystems, whose solution can be written down analytically using generalized inverse matrices. The resulting solution formula is called a generalized split form of a DAE and can be viewed as a certain analogue of the Weierstrass–Kronecker canonical form. In particular, it is shown that arbitrary DAEs with rectangular coefficient matrices are locally reducible to a generalized split form. The structure of these forms (if it is defined on the integration segment) completely determines the structure of general solutions to the systems. DAEs are commonly characterizes by an integer number called index, as well as by the solution space dimension. The dimension of the solution space determines arbitrariness of the the general solution manifold. The index determines how many times we should differentiate the entries on which the solution to the problem depends. We show the ways of calculating these main characteristics.