Abstract
Let be an arbitrary square matrix, an eigenvalue of it, and two systems of linearly independent vectors. A representation of the matrix of scalar resolvents, with th entry equal by definition to , in the form of the product of three matrices , and is obtained, only one of which, , depends on and is a rational function of . On the basis of this factorization and the Binet-Cauchy formula a method for finding the principal part of the Laurent series at the point for the determinant of the matrix of scalar resolvents is put forward and the first two coefficients of the series are found. In the case when at least one of them is distinct from zero, the change after the transition from to of the part of the Jordan normal form corresponding to is determined, where is the operator of rank associated with the systems of vectors and ; and the Jordan basis for the corresponding root subspace of is constructed from Jordan chains of .
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