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Abstract For certain classes of structured matrices, the eigenvalues can be estimated or even calculated explicitly by means of well-known formulas. Starting from these formulas, in this article we derive further formulas for the eigenvalues and singular values of structured matrices that result from second-order linear ODEs by order reduction. This insight leads to a better understanding of these relationships and permits the construction of large-scaled examples (or counterexamples) with specific characteristics that can be used for testing and teaching purposes. As applications, we discuss two well-known example classes, namely a simple discretized wave equation and simple spring-damper systems in series. On the one hand, we can recognize when complex eigenvalues appear and compute the stiffness index and stiffness ratio explicitly. On the other hand, if the dynamic mode decomposition (DMD) is applied to data generated with such ODEs, then the formulas allow to estimate the eigenvalues and singular values that are relevant in this context. For some of the mentioned examples we visualize and discuss the approximations obtained with the DMD with very low-dimensional dynamics.
Diana Estévez Schwarz (Fri,) studied this question.