Given a linear operator in a finite-dimensional complex vectorspace. It is of interest to know how the eigenvalues change with the operator, when the operator depends on a parameter analytically. To answer this question the resultant of the operator is studied in detail. Various properties of the resolvent are stated and proved and these are used to investigate the singularities of the resolvent. Also the partial fraction decomposition of the resolvent has been derived. To this end a number of tools from complex analysis proves useful and some of them are discussed and proved. Since the characteristic equation of the operator, in this finite-dimensional setting, is given by a polynomial equation in two variables, a selection of results on polynomials are stated and proved, so that the characteristic equation can be examined. Algebraic functions are introduced in order to state and prove the main theorem about the behaviour of the eigenvalues. The concept of a norm is used to define the norm of an operator, which is the basic tool in the estimates of the following sections on the resolvent. Finally a number of examples are given to illustrate the theory.