On perturbations of matrix pencils with real spectra. (English) Zbl 0795.15012
Perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil \(A-\alpha B\) 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-Lidskij 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.
Our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of Hermitian matrices such as the Weyl-Lidskij 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.
Reviewer: H.Hollatz (Magdeburg)
MSC:
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15A22 | Matrix pencils |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
Keywords:
perturbation bounds; Davis-Kahan theorems; generalized eigenvalue problem; diagonalizable matrix pencil; real spectrum; generalized eigenspaces; spectral variations; Weyl-Lidskij theorem; Hoffman-Wielandt theorem; Hermitian matricesReferences:
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