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On the norm of normal matrices [r-libre/2603]

Bouthat, Ludovick; Mashreghi, Javad, & Morneau-Guérin, Frédéric (2023). On the norm of normal matrices. RIMS Kôkyûroku Bessatsu 数理解析研究所講究録別冊, 93, 183-222.

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Content : Published Version
Item Type: Journal Articles
Refereed: Yes
Status: Published
Abstract: In this article, we present some recent results related to the calculation of the induced p-norm of n times n circulant matrices A(n, a, b) with diagonal entries equal to a in R and off-diagonal entries equal to b in R. For circulant matrices with nonnegative entries, an explicit formula for the induced p-norm (p between 1 and infinity) is given, whereas for A(n, a, b), a > 0 the situation is no longer so simple and calls for a more subtle analysis. As a matter of fact, while the 2-norm of A(n, a, b) is precisely determined, the exact value of the induced p-norm for 1 < p < infinity and p not equal to 2, still remains elusive. Nevertheless, we provide a lower bound as well as two different categories of upper bounds. As an indication of not being far from the exact values, our estimates coincide at both ends points (i.e., p = 1 and p = infinity) as well as at p = 2 with the precise values. As an abstract approach, we also introduce the *-algebra generated by a normal matrix A accompanied by an axis-oriented norm, and obtain some estimations of the norm of elements of the *-algebra. We then exhibit the connection between the new generalized estimates and the previously obtained estimates in the special case where A is a circulant matrix. Finally, using an optimization-oriented approach, we gain insights on the nature of the maximizing vectors for the p-norm of Ax divided by the p-norm of x. This leads us to formulate a conjecture that, if proven valid, would make it possible to derive an exact formula for the induced p-norm of A(n, a, b) in a particular case of special interest, namely when a = 1-n/n and b = 1/n.
Official URL: https://repository.kulib.kyoto-u.ac.jp/dspace/hand...
Depositor: Morneau-Guérin, Frédéric
Owner / Manager: Frédéric Morneau-Guérin
Deposited: 29 Mar 2022 19:52
Last Modified: 06 Sep 2023 14:08

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