On the Lidskii-Mirsky-Wielandt Theorem, preprint, available at httpwww.wm.eduCASMINEQmprepr.pdfVIP

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2017-04-05 发布于江苏
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On the Lidskii-Mirsky-Wielandt Theorem, preprint, available at httpwww.wm.eduCASMINEQmprepr.pdf

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On the Lidskii-Mirsky-Wielandt Theorem, preprint, available at httpwww.wm.eduCASMINEQmprepr

The Lidskii-Mirsky-Wielandt Theorem {Additive and Multiplicative VersionsChi-Kwong Li and Roy MathiasAbstractWe use a simple matrix splitting technique to give an elementary new proof ofthe Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of theLidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in thestudy of relative perturbation theory for eigenvalues of Hermitian matrices and singularvalues of general matrices. We apply our bound to obtain numerous bounds on thematching distance between the eigenvalues and singular values of matrices. Our resultsstrengthen and generalize those in the literature.1 IntroductionGiven an n n Hermitian matrix A let 1(A) 2(A) n(A) denote its orderedeigenvalues. The singular values of an m n matrix A, are de ned byi(A) = qi(AA); i = 1; 2; : : : ; q minfm;ng:An m n matrix can have at most q minfm;ng non-zero singular values so we shall onlyconsider its largest q singular values.The Lidskii-Mirsky-Wielandt theorem (e.g., see [19, Theorem IV.4.8] or [2, Theorem 9.4])states:Theorem 1.1 Let A and E be n n Hermitian matrices. Then for any indices 1 i1 i2 ik n we have kXj=1[ij (A+ E) ij (A)] kXj=1 j(E): (1.1)This is a very useful result in matrix theory. It is a majorization relation that implies abound on the matching distance between the eigenvalues of A and those of A+ E, namely,kXj=1 jij(A+ E) ij (A)j kXj=1 j(E): (1.2)Department of Mathematics, The College of William and Mary, Williamsburg, Virginia 23187, USA.E-mail: ckli@math.wm.edu, mathias@math.wm.edu. Both authors were supported by grants from the Na-tional Science Foundation. 1 This result can also be stated in terms of norms. A norm kk on Rn is called a symmetricnorm if it is both permutation invariant, that is,kPxk = kxk; 8 x 2 Rn; permutation matrices Pand absolute, that is, k(xi)ni=1k = k(jxij)ni=1k; 8 x 2 Rn:Symmetric norms are also sometimes called symmetric gauge functions. For ev