Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Constant Coefficients

Abstract

Let A, B be $n \times n$ matrices, f a vector-valued function. A and B may both be singular. The differential equation $Ax' + Bx = f$ is studied utilizing the theory of the Drazin inverse. A closed form for all solutions of the differential equation is given when the equation has unique solutions for consistent initial conditions.