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Inverse eigenvalue problems. (English) Zbl 0358.15007
In this paper the author describes two general methods to solve various inverse eigenvalue problems (i.e.p.). The first method is to state an i.e.p. as a system of polynomial equations. By rediscovering the non-linear alternative due to E. Noether and B. L. van der Waerden [Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse, 77–87 (1928; JFM 54.0140.05)] the author shows that the two classical i.e.p. are always solvable with a finite number of solutions over \(C\). [These results were obtained by the author earlier by different methods [Isr. J. Math. 11, 184–189 (1972; Zbl 0252.15004); Linear Algebra Appl. 12, 127–137 (1975; Zbl 0329.15003)]. Most of the paper is devoted to the study of the i.e.p. for symmetric matrices. In that case one looks for real valued solutions which may not exist in general. The crucial step is to reformulate the i.e.p. in such a way that it will have always a real valued solution which will coincide with the original solution, in case that the original solution is solvable over \(R\).
More precisely, \(A^*\) is a solution of \(\min \sum_{i=1}^n(\lambda_i(A)-\omega_i)^2, A \in D\), where \(D\) is a closed set of \(n\times n\) symmetric matrices, \(\{\lambda_i(A)\}_1^n\) is the spectrum of \(A\), \(\{\omega_i\}_1^n\) is the prescribed spectrum and all the sequences are decreasing. By using the maximal characterization of \(\sum_{i=1}^k\lambda_i(A)\) due to \(K\). Fan we obtain a general algorithm to compute \(A^*\) in case that \(D\) is convex. This generalizes the results of O. H. Hald [Compt. Sci. Uppsala Univ. Rep. 42 (1972)]. Certain necessary conditions on \(\{\omega_i\}\) are given if the original i.e.p. is solvable over \(R\).
Reviewer: Shmuel Friedland

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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