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The spectral variation of pencils of matrices. (English) Zbl 0593.15013

Let \((\tilde A,\tilde B)\) be a perturbation of the regular \(n\times n\) matrix pair (A,B) with (possibly infinite) eigenvalues \({\tilde \lambda}{}_ 1,{\tilde \lambda}_ 2,...,{\tilde \lambda}_ n\) and \(\lambda_ 1,\lambda_ 2,...,\lambda_ n\), respectively. Let \(v(\lambda,{\tilde\lambda})= | \lambda -{\tilde \lambda}|\) if \(| \lambda | \leq 1\) and \(v(\lambda,{\tilde\lambda})=| \lambda^{-1}-{\tilde \lambda}^{-1}|\), otherwise. Let \(w(\lambda,{\tilde \lambda})= \min\{| \lambda -{\tilde \lambda}|,| \lambda^{-1}-{\tilde \lambda}^{-1}| \}\) and let \(\rho\) (\(\lambda\),\({\tilde \lambda}\)) denote the chordal metric. The spectral variation \(S_{(A,B)}(\tilde A,\tilde B)=\max_{j}\min_{i}v({\tilde \lambda}_ j,\lambda_ i)\) and similar spectral variations can be defined in terms of w and \(\rho\), respectively. The algebraic structure of a regular pencil \(\lambda A+B\) is investigated by associating with it a ”decomposable pair” (X,T) of \(n\times n\) matrices. The authors then derive bounds for the spectral variations in terms of \(\| \tilde A-A\|\) and \(\| \tilde B- B\|\), both for diagonable pairs and for definite hermitian pairs.
Reviewer: F.J.Gaines

MSC:

15A42 Inequalities involving eigenvalues and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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