Abstract
When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix.
For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix.
A numerical example is given.
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References
J. L. Brenner,New root-location theorems for partitioned matrices, SIAM J. Appl. Math. 16, 889–898 (1968).
Ch. Davis and W. M. Kahan,The rotation of eigenvectors by a perturbation — III, Univ. of Toronto Comp. Sc. Tech. Rep. No. 6 (1968).
D. Feingold and R. Varga,Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Pac. J. Math. 12, 1241–1250 (1962).
G. H. Golub and C. Reinsch,Singular value decomposition and least squares solutions, Handbook series linear algebra, Num. Math. 14, 403–420 (1970).
A. Horn,On the singular values of a product of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 36, 374–375 (1950).
T. Kato,Perturbation theory for linear operators, Springer Verlag, Berlin · Heidelberg · New York, 1966.
H. I. Medley and R. S. Varga,On smallest isolated Gerschgorin disks for eigenvalues III, Num. Math. 11, 361–369 (1968).
A. Ruhe,Properties of a matrix with a very ill-conditioned eigenproblem, Num. Math. 15, 57–60 (1970).
A. Ruhe,An algorithm for numerical determination of the structure of a general matrix, BIT 10, 196–216 (1970).
J. M. Varah,The computation of bounds for the invariant subspaces of a general matrix operator, Stanford Tech. Rep. CS 66 (1967).
J. M. Varah,Computing invariant subspaces of a general matrix when the eigensystem is poorly conditioned, Math. Comp. 24, 137–149 (1970).
J. H. Wilkinson,The algebraic eigenvalue problem, Clarendon Press, Oxford 1965.
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Ruhe, A. Perturbation bounds for means of eigenvalues and invariant subspaces. BIT 10, 343–354 (1970). https://doi.org/10.1007/BF01934203
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DOI: https://doi.org/10.1007/BF01934203