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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Short normal paths and spectral variation

Authors: Rajendra Bhatia and John A. R. Holbrook
Journal: Proc. Amer. Math. Soc. 94 (1985), 377-382
MSC: Primary 15A42
MathSciNet review: 787876
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Abstract: We introduce the notion of a "short normal path" between matrices $ S$ and $ T$, that is, a continuous path from $ S$ to $ T$ consisting of normal matrices and having the same length as the straight line path from $ S$ to $ T$. By this means we prove that for certain normal matrices $ S$ and $ T$ the eigenvalues of $ S$ and $ T$ may be paired in such a way that the maximum distance (in the complex plane) between the pairs is no more than the operator norm $ \left\Vert {S - T} \right\Vert$. In particular, we generalize and provide a new approach to a recent result of Bhatia and Davis treating the case of unitary $ S$ and $ T$.

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Keywords: Matrices, normal operator, spectrum, eigenvalue, spectral variation
Article copyright: © Copyright 1985 American Mathematical Society