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Transactions of the American Mathematical Society

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Analysis of spectral variation and some inequalities


Author: Rajendra Bhatia
Journal: Trans. Amer. Math. Soc. 272 (1982), 323-331
MSC: Primary 15A42; Secondary 15A60, 53A45
DOI: https://doi.org/10.1090/S0002-9947-1982-0656492-X
MathSciNet review: 656492
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Abstract: A geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix. When adapted to special cases, this leads to some classical inequalities as well as some new ones. As an example of the latter, we show that if $ U$, $ V$ are unitary matrices and $ K$ is a skew-Hermitian matrix such that $ U{V^{ - 1}} = \exp K$, then for every unitary-invariant norm the distance between the eigenvalues of $ U$ and those of $ V$ is bounded by $ \vert\vert K\vert\vert$. This generalises two earlier results which used particular unitary-invariant norms.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0656492-X
Keywords: Unitary-invariant norm, eigenvalue, singular value, submanifold, tangent space, $ {C^1}$ functions
Article copyright: © Copyright 1982 American Mathematical Society

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