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

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

MathSciNet review:
787876

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Abstract: We introduce the notion of a "short normal path" between matrices and , that is, a continuous path from to consisting of normal matrices and having the same length as the straight line path from to . By this means we prove that for certain normal matrices and the eigenvalues of and 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 . In particular, we generalize and provide a new approach to a recent result of Bhatia and Davis treating the case of unitary and .

**[1]**R. Bhatia,*Analysis of spectral variation and some inequalities*, Trans. Amer. Math. Soc.**272**(1982), 323-331. MR**656492 (83k:15015)****[2]**R. Bhatia and Ch. Davis,*A bound for the spectral variation of a unitary operator*, Linear and Multilinear Algebra**15**(1984), 71-76. MR**731677 (85b:15020)****[3]**T. Kato,*Perturbation theory for linear operators*, Springer-Verlag, Berlin and New York, 1980.**[4]**L. Mirsky,*Symmetric gauge functions and unitarily invariant norms*, Quart. J. Math. Oxford Ser. (2)**11**(1960), 50-59. MR**0114821 (22:5639)****[5]**V. S. Sunder,*Distance between normal operators*, Proc. Amer. Math. Soc.**84**(1982), 483-484. MR**643734 (83c:47040)****[6]**H. Weyl,*Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen*, Math. Ann.**71**(1912), 441-479. MR**1511670**

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Additional Information

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

Keywords:
Matrices,
normal operator,
spectrum,
eigenvalue,
spectral variation

Article copyright:
© Copyright 1985
American Mathematical Society