Factorization of the Loewner matrix pencil and its consequences

Abstract: This paper starts by deriving a factorization of the Loewner matrix pencil that appears in the data-driven modeling approach known as the Loewner framework and explores its consequences. The first is that the associated quadruple constructed from the data yields a model without requiring further processing. The second consequence is related to how sensitive the eigenvalues of the Loewner pencil are to perturbations. Based on an explicit generalized eigenvalue decomposition of this pencil and by making use of perturbation theory of matrix pencils, we explore two types of eigenvalue sensitivities. The first one is defined with respect to unstructured perturbations of the Loewner pencil, while the second one is defined for structured perturbations. We also discuss how the choice of data affects the two sensitivities.