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On the variation of the spectrum of a normal matrix
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
1996 (English)In: Linear Algebra and Its Applications, Vol. 246, 215-223 p.Article in journal (Refereed) Published
Abstract [en]

Let A and (A) over tilde be two n x n normal matrices with spectra {lambda(j)} and {<(lambda)over tilde (j)>}. Then by the Hoffman-Wielandt theorem, there is a permutation pi of (1,...,n) such that root(n) Sigma(j = 1) \<(lambda)over tilde (pi(j))> - lambda(j)\(2) less than or equal to parallel to (A) over tilde - A parallel to(F), where parallel to parallel to(F) denotes the Frobenius norm. However, if A is normal but (A) over tilde nonnormal, it may be asked: How to relate the eigenvalues of (A) over tilde to those of A? An answer is given in this paper: There is a permutation pi of {1, 2,...,n} such that root(n) Sigma(j = 1) \<(lambda)over tilde (pi(j))> - lambda(j)\(2) less than or equal to root n parallel to (A) over tilde - A parallel to(F), and the factor root n is best possible. As a corollary, we have max (j) \<(lambda)over tilde (pi(j))> - lambda(j)\ less than or equal to n parallel to (A) over tilde - A parallel to(2).

Place, publisher, year, edition, pages
1996. Vol. 246, 215-223 p.
National Category
Computer Science
Research subject
Numerical Analysis
Identifiers
URN: urn:nbn:se:umu:diva-20763ISBN: 0024-3795 OAI: oai:DiVA.org:umu-20763DiVA: diva2:209448
Available from: 2009-03-25 Created: 2009-03-25 Last updated: 2009-03-25

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