A constrained eigenvalue problem. (English) Zbl 0666.15006

For the problem of finding \(Min(x^ TAx)\) for a symmetric matrix A subject to \(x^ Tx=1\) and \(N^ Tx=t\) theoretical and numerical methods are described. First the linear constraint is removed and then Lagrange multipliers are employed reducing the problem to solve either a secular equation or a quadratic eigenvalue problem.
Reviewer: V.Mehrmann


15A18 Eigenvalues, singular values, and eigenvectors
65F15 Numerical computation of eigenvalues and eigenvectors of matrices


Full Text: DOI Link


[1] Draper, N.R., “ridge analysis” of response surfaces, Technometrics, 5, 469-479, (1963) · Zbl 0124.35003
[2] Forsythe, G.E.; Golub, G.H., On the stationary values of a second-degree polynomial on the unit sphere, SIAM J. appl. math., 13, 1050-1068, (1963) · Zbl 0168.03005
[3] Gander, W., Least squares with a quadratic constraint, Numer. math., 36, 291-307, (1981) · Zbl 0437.65031
[4] Golub, G.H., Some modified matrix eigenvalue problems, SIAM rev., 15, 318-334, (1973) · Zbl 0254.65027
[5] Golub, G.H.; Heath, M.; Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21, 215-223, (1979) · Zbl 0461.62059
[6] Golub, G.H.; Van Loan, C.F., Matrix computations, (1983), John Hopkins U.P Baltimore · Zbl 0559.65011
[7] Moré, J.J., The Levenberg-Marquardt algorithm: implementation and theory, (), 105-116 · Zbl 0372.65022
[8] Moré, J.J.; Sorensen, D.C., Computing a trust region step, SIAM J. sci. statist. comput., 4, 553-572, (1983) · Zbl 0551.65042
[9] Reinsch, Chr.H., Smoothing by spline functions, II, Numer. math., 16, 451-454, (1971) · Zbl 1248.65020
[10] Spjøtvoll, E., A note on a theorem of forsythe and Golub, SIAM J. appl. math., 23, 307-311, (1972) · Zbl 0259.15019
[11] Spjøtvoll, E., Multiple comparison of regression functions, Ann. math. statist., 43, 1076-1088, (1972) · Zbl 0241.62046
[12] von Matt, U., A constrained eigenvalue problem, () · Zbl 0742.65044
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.