Inverse Eigenvalue Probelms: Theory and Applications

 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.

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