Structured Inverse Eigenvalue Problems

An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behavior. Spectral information is entailed because the dynamical behavior often is governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system often is subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure the matrices embodied can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well the given spectral property. In this expository paper the emphasis is to provide an overview of the vast scope of this intriguing problem, treating some of its many applications, its mathematical properties, and a variety of numerical techniques.