### Resolvent of Matrix Polynomials, Pseudospectra and Inversion Problems

#### Abstract

This paper deals with the mathematical model of a dynamic systems written by the n ordinary differential equations of order m with constant coefficients. The formulas for the resolvent of matrix polynomial (l-matrix problem) and for "derivative resolvents", as well theirs derivations with respect to l are given with help of less commonly used linearization form. The extension of Pseudospectra definition for above matrix polynomial problem is proposed. The inversion problem, formulated here for general complex and real coefficient matrices is important in numerical simulation. As a motivation of the study of an evolutive dynamic system depending on a one parameter, the famous Lancaster's example, where m=2 and n=4, is given.

#### Keywords

Dynamic systems; regular matrix polynomial problem; linearization; eigenvalue; eigenvector; latent roots; latent vector; generalized latent vector; Jordan canonical form; resolvent; state space; component matrices; pseudospectra; inversion problem.