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On a conjecture on the closest normal matrix
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
1998 (English)In: Mathematical Inequalities & Applications, Vol. 1, no 3, 305-318 p.Article in journal (Refereed) Published
Abstract [en]

Let A be a complex n x n matrix and let N-n be the set of normal n x n matrices. A conjecture is that parallel to A - N(n)parallel to(F)(2) less than or equal to n - 1/ndep(2)(A), where dep(2)(A) = parallel to A parallel to(F)(2) - Sigma(i=1)(n) lambda(i)(2)(A) and lambda(i)(A), i = 1,...,n are the eigenvalues of A. We prove that the conjecture is correct for all even n and for n = 3, 5, 7. However, for the dimensions, n = 3, 5, 6, 7, and presumably also other problem dimensions it is possible to derive sharper bounds. We also prove a bound for odd n which converges to the bound in the conjecture when n tends to infinity. The main idea in the proofs is to use LP problems with constraints based on different ways to approximate A with normal matrices.

Place, publisher, year, edition, pages
1998. Vol. 1, no 3, 305-318 p.
URN: urn:nbn:se:umu:diva-21523ISBN: 1331-4343OAI: diva2:211371
Available from: 2009-04-14 Created: 2009-04-14 Last updated: 2009-04-14

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