Structured inverse eigenvalue problems

Abstract

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 behaviour. Spectral information is entailed because the dynamical behaviour is often governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system is often subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure embodied by the matrices can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well as 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.

An inexact Cayley transform method for inverse eigenvalue problems

“…When the systems are large, we may reduce the computational cost by solving both systems iteratively. One could expect that it requires only a few iterations to solve (10) iteratively. This is due to the fact that, as {c k } converges to c * , Z k converges to zero, see [17,Equation (3.64)].…”

“…To guarantee the orthogonality of Q k in (10) and P k in (15), both systems are solved up to machine precision eps (which is ≈ 2.2×10 −16 ). We use the right-hand side vector as the initial guess for these two systems.…”

“…As mentioned in § §2-3, solving the linear systems (10) and (15) iteratively will require only a few iterations since the coefficient matrices of these systems converge to the identity matrix as c k converges to c * . We report in Table 3 the numbers of iterations required for convergence for these systems, averaged over the ten test problems with n = 100 and 200.…”

“…A good reference for these applications is the recent survey paper on structured inverse eigenvalue problems by Chu and Golub [10]. In many of these applications, the problem size n can be large.…”

An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

“…They also occur in a remarkable variety of applications [2,[5][6][7][8] and have been studied for different classes of structured matrices. We refer the reader to [9][10][11][12] and references therein.…”