Abstract
Let\(\hat \lambda \) and\(\hat x\) be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in\(\hat \lambda \) and\(\hat \lambda \). We present one absolute perturbation bound and two relative perturbation bounds.
The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that\(\hat \lambda \) and\(\hat x\) are an exact eigenpair of a perturbed matrixD 1 AD 2 , whereD 1 andD 2 are non-singular, butD 1 AD 2 is not necessarily diagonalisable. We derive a bound on the relative error in\(\hat \lambda \) and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD 1 andD 2 from similarity and the deviation ofD 2 from identity.
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Communicated by Axel Ruhe.
This work was partially supported by NSF grant CCR-9400921.
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Eisenstat, S.C., Ipsen, I.C.F. Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices. Bit Numer Math 38, 502–509 (1998). https://doi.org/10.1007/BF02510256
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DOI: https://doi.org/10.1007/BF02510256