In this thesis, we derive a factorization of the Loewner pencil in data-driven modeling and explore its consequences. The Loewner framework is a data-driven modeling and complexity reduction method that can be used to learn models of dynamical systems from measurements of their transfer function. One key feature of the Loewner framework consists in the fact that it does not need an exact description of the original dynamical system to start with, which is typically described by ordinary or partial differential equations (ODEs, PDEs). Instead of having full access to the coefficient matrices that scale these equations, one requires only transfer function measurement values. Finally, by arranging the given data in a specific way, one can construct with basically no computational effort a realization (dynamical system) that explains the data. The Loewner pencil plays a central role in the system realization constructed by the Loewner framework. More precisely, the two Loewner matrices that enter the pencil represent the coefficient matrices that scale the internal variable vector and its derivative. Consequently, the eigenvalues of the pencil are the poles of the surrogate Loewner model and are used to characterize the dynamics of the system.

In this thesis, the Loewner pencil is factorized in terms of generalized Cauchy matrices that are composed of poles, residues of the system, and measurement points. It is shown that the factors given by the generalized Cauchy matrices are Krylov projection matrices for a particular system realization. Using the factorization of the generalized Loewner matrix, the eigenvalue decomposition (EVD) of the Loewner pencil is hence available. Based on this EVD and eigenvalue perturbation theory for matrix pencils, we explore two types of eigenvalue sensitivities. The first one is defined for unstructured perturbations of the Loewner pencil, while the second one is defined for structured perturbations. The motivation for studying these two sensitivities is that they reflect the robustness of the Loewner surrogate model. We will show that the unstructured perturbation sensitivity is related to the numerical conditioning of the Loewner pencil and can be used in comparison to the pseudo spectrum of the pencil. Moreover, it is shown that the structured perturbation sensitivity can be used to estimate eigenvalue perturbations as a result of the noise in the data. We also discuss how the choice of data affects the two sensitivities. Finally, we will extend our framework to the time-series data and show its application in the research of biological rhythms.