Short normal paths and spectral variation

Bhatia, Rajendra; Holbrook, John A. R.

doi:10.1090/S0002-9939-1985-0787876-5

Short normal paths and spectral variation
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by Rajendra Bhatia and John A. R. Holbrook

Proc. Amer. Math. Soc. 94 (1985), 377-382

DOI: https://doi.org/10.1090/S0002-9939-1985-0787876-5

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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 $\left \| {S - T} \right \|$. 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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