zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Anderson, J.; Parker, J., Choices for A in the matrix equation T=AB-BA, Linear and multilinear algebra, 2, 203-209, (1974)
[2] Borg, G., Eine umkehrung der Sturm-liouvilleschen eigenwertaufgabe, Acta math., 78, 1-96, (1946) · Zbl 0063.00523
[3] Dunford, N.; Schwartz, J.T., Linear operators, parts I and II, (1958 and 1963), Interscience New York
[4] Fan, K., On a theorem of Weyl concerning eigenvalues of linear transformations, Proc. natl. acad. sci., 35, 652-655, (1949)
[5] Friedland, S., Matrices with prescribed off-diagonal elements, Isr. J. math., 11, 184-189, (1972) · Zbl 0252.15004
[6] Friedland, S., Extremal eigenvalue problems for convex sets of symmetric matrices and operators, Isr. J. math., 15, 311-331, (1973) · Zbl 0279.15007
[7] Friedland, S., On inverse multiplicative eigenvalue problems for matrices, Linear algebra appl., 12, 127-137, (1975) · Zbl 0329.15003
[8] Friedland, S.; Karlin, S., Some inequalities for the spectral radius of non-negative matrices and applications, Duke math. J., 42, 459-490, (1975) · Zbl 0373.15008
[9] Gantmacher, F.R.; Krein, M.G., Oscillating matrices and kernels and small vibrations of mechanical systems, (1950), Gos. Izd Moscow · Zbl 0041.35502
[10] Gunning, R.C.; Rossi, H., Analytic functions of several complex, (1965), Prentice-Hall Englewood, N.J · Zbl 0141.08601
[11] Hadeler, K.P., Ein inverses eigenwertproblem, Linear algebra appl., 1, 83-101, (1968) · Zbl 0159.03602
[12] Hadeler, K.P., Multiplikative inverse eigenwertprobleme, Linear algebra appl., 2, 65-86, (1969) · Zbl 0182.05201
[13] Hald, O.H., On discrete and numerical inverse strum-Liouville problems, (1972), Dept. Comput. Sci., Uppsala Univ, Rep. No. 42
[14] Hochstadt, H., On the construction of a Jacobi matrix from spectral data, Linear algebra appl., 8, 435-446, (1974) · Zbl 0288.15029
[15] Hoffman, A.J.; Wielandt, H.W., The variation of the spectrum of a normal matrix, Duke math. J., 20, 37-39, (1953) · Zbl 0051.00903
[16] Ince, E.L., Ordinary differential equations, (1956), Dover New York · Zbl 0063.02971
[17] Johnson, C.R., A note on matrix solutions to A=XY-YX, Proc. am. math. soc., 42, 351-353, (1974) · Zbl 0278.15007
[18] Laborde, F., Sur un problème inverse de valeurs propres, C.R. acad. sci. Paris, 268, A-153-A-156, (1969) · Zbl 0174.47001
[19] Levitan, B.M., Generalized translation operators, (1964), Israel Program for Sci. Transl Jerusalem · Zbl 0192.48902
[20] Marcus, M.; Minc, H., A survey of matrix theory and matrix inequalities, (1964), Prindle, Weber and Schmidt · Zbl 0126.02404
[21] Morel, P., A propos d’un problème inverse de valeurs propres, C.R. acad. sci. Paris, 277, A-125-A-128, (1973) · Zbl 0262.65028
[22] Morel, P., Sur le problème inverse de valeurs propres, Numer. math, 23, 83-94, (1974) · Zbl 0353.15017
[23] Neumark, M.A., Lineare differential operatoren, (1963), Akademie Berlin · Zbl 0114.28703
[24] de Oliveira, G.N., Note on an inverse characteristic value problem, Numer. math., 15, 345-347, (1970) · Zbl 0202.03504
[25] de Oliveira, G.N., Matrix inequalities and the additive inverse eigenvalue problem, Computing, 9, 95-100, (1972) · Zbl 0247.15007
[26] de Oliveira, G.N., On the multiplicative inverse eigenvalue problem, Can. math. bull., 15, 189-193, (1972) · Zbl 0247.15006
[27] Pólya, G.; Schiffer, M., Convexity of functionals by transplantation, J. anal. math., 3, 245-345, (1953/4) · Zbl 0056.32701
[28] Van der Waerden, B.L., Modern algebra, Vol. II, (1950), Ungar New York · Zbl 0037.01903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.