Laurent expansion for the determinant of the matrix of scalar resolvents

Let $A$ be an arbitrary square matrix, $\lambda$ an eigenvalue of it, $\{\xi_1,\dots,\xi_r\}$ and $\{\eta_1,\dots,\eta_r\}$ two systems of linearly independent vectors. A representation of the matrix of scalar resolvents, with $ij$th entry equal by definition to $(\xi_i,(zE-A)^{-1}\eta_j)$, in the form of the product of three matrices $\Xi$, $\Delta(z)$, and $\Psi^T$ is obtained, only one of which, $\Delta(z)$, depends on $z$ and is a rational function of $z$. 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 $z=\lambda$ 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 $A$ to $A+B$ of the part of the Jordan normal form corresponding to $\lambda$ is determined, where $B=\sum_{i=1}^r(\,\cdot\,,\xi_i)\eta_i$ is the operator of rank $r$ associated with the systems of vectors $\{\xi_1,\dots,\xi_r\}$ and $\{\eta_1,\dots,\eta_r\}$; and the Jordan basis for the corresponding root subspace of $A+B$ is constructed from Jordan chains of $A$.