Abstract
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.
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Ruhe, A. An algorithm for numerical determination of the structure of a general matrix. BIT 10, 196–216 (1970). https://doi.org/10.1007/BF01936867
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DOI: https://doi.org/10.1007/BF01936867