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2011.114: The tan theta theorem with relaxed conditions

2011.114: Yuji Nakatsukasa (2011) The tan theta theorem with relaxed conditions.

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Abstract

The Davis-Kahan tan theta theorem bounds the tangent of the angles between an approximate and an exact invariant subspace of a Hermitian matrix. When applicable, it gives a sharper bound than the sin theta theorem. However, the tan theta theorem requires more restrictive conditions on the spectrums, demanding that the entire approximate eigenvalues (Ritz values) lie above (or below) the set of exact eigenvalues corresponding to the orthogonal complement of the invariant subspace. In this paper we show that the conditions of the tan theta theorem can be relaxed, in that the same bound holds even when the Ritz values lie both below and above the exact eigenvalues, but not vice versa.

Item Type:MIMS Preprint
Uncontrolled Keywords:Davis-Kahan, tan theta theorem, sin theta theorem, generalized tan theorem, eigenvector
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2011.114
Deposited By:Yuji Nakatsukasa
Deposited On:19 December 2011

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