The Laurent Expansion of a Generalized Resolvent with Some Applications

Abstract

Let A, B be $n \times n$ complex matrices and assume $(A + \lambda B)^{ - 1} $ exists for some complex number $\lambda $; then $(A + \lambda B)^{ - 1} $ has a Laurent expansion of the form $\sum_{k = - \nu }^\infty {Q_k \lambda ^k } $ with $Q_{ - \nu } \ne 0$ valid in some deleted neighborhood of $\lambda = 0$. Explicit formulas for the $Q_k $ are given. Expansions of powers of $(A + \lambda B)^{ - 1} A$ are also given. The expansions are used to obtain various characterizations of the Drazin inverse and other inverses as limits. Finally the expansions, together with Laplace transforms, are used to solve the differential equations $A\dot x + Bx = 0$ and $A\ddot x + Bx = 0$, where A may be singular, in the case when unique solutions exist for appropriate initial conditions.