Perturbations of the Natural Frequencies and Eigenvectors of a Network

A basic problem in network theory is the determination of the displacements of the natural frequencies of a system, due to small variations in its elements. This problem is usually solved by evaluating the effect of the variation of each element. In the following, the unknown displacements are determined directly in terms of the variation of all elements. The analysis includes the regular case with simple and multiple roots, and the singular case resulting from an increase of the order of the system function.