An inverse eigenvalue problem concerns the reconstruction
of a matrix from prescribed spectral data.
The spectral data involved may consist of the complete or only partial information of eigenvalues or
eigenvectors. The objective of an inverse eigenvalue problem is to construct a matrix that maintains a
certain specific structure as well as that given spectral property.
Inverse eigenvalue problems arise in a remarkable variety of applications,
including system and control
theory, geophysics, molecular spectroscopy, particle physics, structure analysis, and so on. Generally
speaking, the basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of
a certain system from the knowledge or desire of its dynamical behavior. Since the dynamical behavior
often is governed by the underlying natural frequencies and/or normal modes, the spectral constraints
are thus imposed. On the other hand, in order that the resulting model is physically realizable, additional
structural constraints must also be imposed upon the matrix. Depending on the application, inverse
eigenvalue problems appear in many different forms.
Associated with any inverse eigenvalue problem are two fundamental
questions -- the theoretic issue on
solvability and the practical issue on computability. Solvability concerns obtaining a necessary or a sufficient
condition under which an inverse eigenvalue problem has a solution. Computability concerns developing a
procedure by which, knowing a priori that the given spectral data are feasible, a matrix can be constructed
numerically. Both questions are difficult and challenging.
In this note the emphasis is to provide an overview of the vast
scope of this fascinating problem. The fundamental
questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as
well as several open problems will be discussed.
This research was supported in part by the National
Science Foundation, under the grants DMS-9803759 and
DMS-0073056. The lectures are to be presented at the Istituto per Ricerche di Matematica Applicata from
June 23 to July 20, 2001, upon the invitation of Fasma Diele. The visit to present this series of lectures is made
possible by Professor Roberto Peluso at the IRMA and Professor Dario Bini at the Universita' di Pisa with the
support from Il Consiglio Nazionale delle Ricerche (CNR) and the Gruppo Nazionale per il Calcolo Scientifico
(GNCS) of the Istituto Nazionale di Alta Matematica (INDAM) under the project "Algebra Lineare Numerica
per Problemi con Struttura e Applicazioni". The warm kindness and encouragement received from these colleagues
are greatly appreciated.