The reconstruction of a specially structured Jacobi matrix with an application to damage detection in rods

A finite-element discretization of the differential equation for the axial vibration of a rod with varying cross-section leads to a specially structured n × n matrix pencil. The reconstruction of this pencil from its spectrum can be achieved by the reconstruction of a unique Jacobi matrix using half of its spectrum and half of the spectrum of its principal submatrix of order (n − 1). The technique is used in an optimization problem formulated for damage detection in rods defined in terms of changes in the effective cross-sectional area.