An algorithm for numerical determination of the structure of a general matrix

An algorithm, proposed by V. N. Kublanovskaya, for solving the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix, is studied theoretically and numerically. It uses successiveQR-factorizations to determine annihilated subspaces.An adaptation of the algorithm is developed which, applied to a matrix with a very ill-conditioned eigenproblem, computes a degenerate matrix. The difference between these matrices is small, measured in the spectral norm. The degenerate matrix will appear in a standard form, whose eigenvalues and principal vectors can be computed in a numerically stable manner.Numerical examples are given.

[1]  K. Fan,et al.  Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[2]  J. G. F. Francis,et al.  The QR Transformation A Unitary Analogue to the LR Transformation - Part 1 , 1961, Comput. J..

[3]  C. Desoer,et al.  A Note on Pseudoinverses , 1963 .

[4]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[5]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[6]  V. Kublanovskaya,et al.  On a method of solving the complete eigenvalue problem for a degenerate matrix , 1966 .

[7]  G. Golub.  Least squares, singular values and matrix approximations , 1968 .

[8]  J. Varah.  The calculation of the eigenvectors of a general complex matrix by inverse iteration , 1968 .

[9]  J. M. Varah.  Rigorous machine bounds for the eigensystem of a general complex matrix , 1968 .

[10]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[11]  Axel Ruhe.  Properties of a matrix with a very ill-conditioned eigenproblem , 1970 .

[12]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[13]  Axel Ruhe.  Perturbation bounds for means of eigenvalues and invariant subspaces , 1970 .

[14]  J. Varah,et al.  Computing Invariant Subspaces of a General Matrix when the Eigensystem is Poorly Conditioned , 1970 .

[15]  James Hardy Wilkinson,et al.  The QR and QL Algorithms for Symmetric Matrices , 1971 .