Analysis of spectral variation and some inequalities

Bhatia, Rajendra

doi:10.1090/S0002-9947-1982-0656492-X

Analysis of spectral variation and some inequalities
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Trans. Amer. Math. Soc. 272 (1982), 323-331 Request permission

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 $||K||$. This generalises two earlier results which used particular unitary-invariant norms.

References

William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
Louis Auslander and Robert E. MacKenzie, Introduction to differentiable manifolds, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1963. MR 0161254
Rajendra Bhatia and Shmuel Friedland, Variation of Grassmann powers and spectra, Linear Algebra Appl. 40 (1981), 1–18. MR 629603, DOI 10.1016/0024-3795(81)90136-1
Rajendra Bhatia and Kalyan K. Mukherjea, On the rate of change of spectra of operators, Linear Algebra Appl. 27 (1979), 147–157. MR 545728, DOI 10.1016/0024-3795(79)90037-5
Chandler Davis, Various averaging operations onto subalgebras, Illinois J. Math. 3 (1959), 538–553. MR 112042
Chandler Davis, The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6 (1963), 159–173. MR 149309, DOI 10.1016/0022-247X(63)90001-5
Chandler Davis, The rotation of eigenvectors by a perturbation. II, J. Math. Anal. Appl. 11 (1965), 20–27. MR 180852, DOI 10.1016/0022-247X(65)90066-1
Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760–766. MR 45952, DOI 10.1073/pnas.37.11.760
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
Peter Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math. 4 (1962), 24–40. MR 135706, DOI 10.1007/BF01386294
A. J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), 37–39. MR 52379
L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford Ser. (2) 11 (1960), 50–59. MR 114821, DOI 10.1093/qmath/11.1.50
Alexander Ostrowski, Über die Stetigkeit von charakteristischen Wurzeln in Abhängigkeit von den Matrizenelementen, Jber. Deutsch. Math.-Verein. 60 (1957), no. Abt. 1, 40–42 (German). MR 89171
K. R. Parthasarathy, Eigenvalues of matrix-valued analytic maps, J. Austral. Math. Soc. Ser. A 26 (1978), no. 2, 179–197. MR 511603
Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112

Some matrix inequalities and metrization of matric space

1

Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479 (German). MR 1511670, DOI 10.1007/BF01456804
Helmut Wielandt, An extremum property of sums of eigenvalues, Proc. Amer. Math. Soc. 6 (1955), 106–110. MR 67842, DOI 10.1090/S0002-9939-1955-0067842-9