On the additive inverse eigenvalue problem
DOI:
https://doi.org/10.22199/S07160917.1988.0014.00001Keywords:
Autovalores, Inversos aditivosAbstract
The problem of determining the eigenvalues of a given matrix A is one of long-standing and wide application in many areas of science and engineering. By contrast, the problem of determining all or some of the entries of A fron spectral information is a new subject which has only recently become an active area of research. We quote Z. Bohte [2], (p. 385, 1967): "For the numerical solution of this problem a number of techniques have been used without sufficient theoretical consideration. We believe the problem has not attracted the attention of the mathematicians yet".References
1. Biegler-Konig, F. W. Sufficient Conditiors for the Solvability of Inverse Eigenvalue Problems. Linear Algebra and Its Applications 40: 89- 100, 1981.
2. Bohte, Z. Numerical Solution of the Inverse Algebraic Eigenvalue Problem. Computer Journal 10: 385-388, 1967/1968.
3. Brussard, P. J., and Glaudemans, P. W. Shell Model Applications in Nuclear Spectroscopy. Elsevier, New York, 1977.
4. De Oliveira, G. N. Note on an Inverse Characteristic Value Problem. Numerische Mathematik 15: 345-347, 1970.
5. Downing, A. C., Jr., and Householder, A. S. Some Inverse Characteristic Value Problems. Journal Assoc. for Computing Machinery 3: 203-207, 1956.
6. Faddeyev, L. D. The Inverse Problem in the Quantun Theory of Scattering. Journal of Mathematical Physics 4: 72-104, 1963.
7. Fletcher, R. Semi-definite Matrix Constraints in Optimization. Siam Journal Control Optimization 23: 493-513, 1985.
8. Friedland, S. lnverse Eigenvalue Problems. Linear Algebra and lts Applications 17: 15-51, 1977.
9. Friedland, S., Nocedal, J., and Overton, M. L. The Formulation and Analysis of Numerical Methods for lnverse Eigenvalue Problems. Siam Journal of Numerical Analysis 24: 634-667, 1987.
10. Hadeler, K. P. Ein Inverses Eigenwertproblem. Linear Algebra and Its Applications 1: 83-101, 1968.
11. Hoffman, A., and Wielandt, H. The Variation of the Spectrum of a Normal Matrix. Duke Math. Journal 2O: 37 - 4O, 1953
12. Kato, T. Perturbation Theory for Linear Operators. Springer-Verlag, New York, Inc., 1966.
13. Laborde F. Sur un Probleme Inverse d'un Probleme de Valeurs Propres. C. R. Acad. Sc. Paris 268: 153-156, 1969.
14. Marcus, M, and Mino, H. A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacor, Boston, 1964.
15. Mirsky, L. The Spread of a Matrix. Mathematics 3: 127-130, 1956.
16. Morel, P. A Propos d'un Probleme Inverse de Valeurs Propres. C. R. Acad. Sc. Paris 277: 125-128, 1973.
17 . Morel, P . Sur le Probleme Inverse des Valeurs Propres. Numerische Mathematik 23: 83-94, 1974.
18. Morel, P. Des Algorithmes pour le Probleme Inverse des Valeurs Propres. Linear Algebra and Its Applications 13: 251-273, 1976.
19. Pliva, J., and Toman, S. Multiplicity of Solutions of the lnverse Secular Problem. Journal of Molecular Spectroscopy 21: 362-371, 1966.
20. Soto, R. L. On Matrix Inverse Eigenvalue Problems. Ph. D. Dissertation. New Mexico State University, 1987.
21. Stewart, G. W. Introduction to Matrix Computation. Academic Press, New York, 1973.
22. Wilkinson, J. H. The Algebraic Eigenvalue Problem. Oxford University Press, London, 1965.
2. Bohte, Z. Numerical Solution of the Inverse Algebraic Eigenvalue Problem. Computer Journal 10: 385-388, 1967/1968.
3. Brussard, P. J., and Glaudemans, P. W. Shell Model Applications in Nuclear Spectroscopy. Elsevier, New York, 1977.
4. De Oliveira, G. N. Note on an Inverse Characteristic Value Problem. Numerische Mathematik 15: 345-347, 1970.
5. Downing, A. C., Jr., and Householder, A. S. Some Inverse Characteristic Value Problems. Journal Assoc. for Computing Machinery 3: 203-207, 1956.
6. Faddeyev, L. D. The Inverse Problem in the Quantun Theory of Scattering. Journal of Mathematical Physics 4: 72-104, 1963.
7. Fletcher, R. Semi-definite Matrix Constraints in Optimization. Siam Journal Control Optimization 23: 493-513, 1985.
8. Friedland, S. lnverse Eigenvalue Problems. Linear Algebra and lts Applications 17: 15-51, 1977.
9. Friedland, S., Nocedal, J., and Overton, M. L. The Formulation and Analysis of Numerical Methods for lnverse Eigenvalue Problems. Siam Journal of Numerical Analysis 24: 634-667, 1987.
10. Hadeler, K. P. Ein Inverses Eigenwertproblem. Linear Algebra and Its Applications 1: 83-101, 1968.
11. Hoffman, A., and Wielandt, H. The Variation of the Spectrum of a Normal Matrix. Duke Math. Journal 2O: 37 - 4O, 1953
12. Kato, T. Perturbation Theory for Linear Operators. Springer-Verlag, New York, Inc., 1966.
13. Laborde F. Sur un Probleme Inverse d'un Probleme de Valeurs Propres. C. R. Acad. Sc. Paris 268: 153-156, 1969.
14. Marcus, M, and Mino, H. A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacor, Boston, 1964.
15. Mirsky, L. The Spread of a Matrix. Mathematics 3: 127-130, 1956.
16. Morel, P. A Propos d'un Probleme Inverse de Valeurs Propres. C. R. Acad. Sc. Paris 277: 125-128, 1973.
17 . Morel, P . Sur le Probleme Inverse des Valeurs Propres. Numerische Mathematik 23: 83-94, 1974.
18. Morel, P. Des Algorithmes pour le Probleme Inverse des Valeurs Propres. Linear Algebra and Its Applications 13: 251-273, 1976.
19. Pliva, J., and Toman, S. Multiplicity of Solutions of the lnverse Secular Problem. Journal of Molecular Spectroscopy 21: 362-371, 1966.
20. Soto, R. L. On Matrix Inverse Eigenvalue Problems. Ph. D. Dissertation. New Mexico State University, 1987.
21. Stewart, G. W. Introduction to Matrix Computation. Academic Press, New York, 1973.
22. Wilkinson, J. H. The Algebraic Eigenvalue Problem. Oxford University Press, London, 1965.
Published
2018-03-28
How to Cite
[1]
Óscar L. Rojo Jeraldo and R. L. Soto Montero, “On the additive inverse eigenvalue problem”, Proyecciones (Antofagasta, On line), vol. 7, no. 14, pp. 1-27, Mar. 2018.
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