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Inverse eigenvalue problems. (English) Zbl 0915.15008
In the inverse eigenvalue problem, one has to construct a matrix with a (partially) given spectrum. The problem appears in many different forms and in many different applications. Usually the problem is constrained in the sense that the matrix \(M\) that one wants to find has to be in a certain class. For example it should be of the form \(M=A+X\) or \(M=AX\) where \(A\) is a given matrix and \(X\) belongs to a certain class of matrices, or \(M\) could be forced to be of a certain structure, like for example to be a Toeplitz or a positive definite or a stochastic matrix. If the problem has no solution, then one may want to solve the best possible solution in some sense. For example using a least squares criterion, we could relax the condition on the spectrum or the condition on the class to which \(M\) should belong.
These (and other) inverse eigenvalue problems are surveyed in this paper. For each problem, the question of solvability and possibly sensitivity of the problem are discussed whenever results are available. Also the numerical techniques to solve these problems are discussed. A non-exhaustive list of about 200 relevant references are given. The paper clearly exposes that although there is a huge literature on these problems, there are still many unanswered questions and many issues that are open research problems.

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
34A55 Inverse problems involving ordinary differential equations
35R30 Inverse problems for PDEs
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