The nature of the eigensolution of a general self-adjoint system in which two eigenvalues become nearly equal is studied for the relationship between the eigenvalue loci, which are the plot of eigenvalues versus a system parameter, and the manner in which the eigenfunctions change with changes in that parameter. A perturbation series solution is used to relate the eigenvalues and eigenfunctions for different values of the parameter. Criteria governing the occurrence of veering of these loci, as opposed to intersection, are established. It is proven that in the veering region, the eigenfunctions are linear combinations of the corresponding eigenfunctions at the outer limits of the veering zone, and that those combinations are highly sensitive to the value of the system parameter. A corollary is that in systems whose individual subcomponents are lightly coupled and nearly identical, eigenvalue veering and high mode sensitivity are associated with mode localization, in which some of the system modes are strongly enhanced in a small portion of the domain. An example of a two-span beam with irregular spacing of the supports, whose exact eigensolution has been shown to feature mode localization and high mode sensitivity, demonstrates the accuracy of the perturbation solution. A rectangular clamped membrane is used to illustrate the relationship between approximation error in the derivation of the system equations and eigenvalue veering.

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