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The paper presents a survey of theory and application of an extended version of Gershgorin's theorems towards infinite matrix representations of distributed parameter systems (DPS). The paper gives an answer to the question: How are the eigenvalues of the actually infinite-dimensional control system related to those of its finite Galerkin approximation? Modal expansions are covered as a special case. Two theorems will be provided based on different assumptions on diagonal dominance. As a result, all eigenvalues of the DPS will be enclosed in Gershgorin disks. This provides a sufficient stability criterion, whenever the spectrum of the DPS consists only of eigenvalues. The basic results will be applied to a number of control problems, including numerical examples.