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Mathematics of Computation

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On perturbations of matrix pencils with real spectra

Author: Ren Cang Li
Journal: Math. Comp. 62 (1994), 231-265
MSC: Primary 15A22; Secondary 65F15, 65F35
MathSciNet review: 1208838
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Abstract: Perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil $ A - \lambda B$ with real spectrum are developed. It is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices. The applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds. Our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of Hermitian matrices such as the Weyl-Lidskii theorem and the Hoffman-Wielandt theorem; and those for generalized eigenspaces are counterparts of the celebrated Davis-Kahan $ \sin \theta ,\sin 2\theta $ theorems for the eigenspace variations of Hermitian matrices.

The paper consists of two parts. Part I is for generalized eigenvalue perturbations, while Part II deals with generalized eigenspace perturbations.

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Keywords: Diagonalizable matrix pencil, definite pencils, real spectrum, perturbation bounds
Article copyright: © Copyright 1994 American Mathematical Society