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Bouthat, Ludovick; Mashreghi, Javad, & MorneauGuérin, Frédéric (2022). On the norm of normal matrices. RIMS Kôkyûroku Bessatsu.
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Content : Accepted 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 pnorm of n times n circulant matrices A(n, a, b) with diagonal entries equal to a in R and offdiagonal entries equal to b in R. For circulant matrices with nonnegative entries, an explicit formula for the induced pnorm (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 2norm of A(n, a, b) is precisely determined, the exact value of the induced pnorm 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 axisoriented 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 optimizationoriented approach, we gain insights on the nature of the maximizing vectors for the pnorm of Ax divided by the pnorm 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 pnorm of A(n, a, b) in a particular case of special interest, namely when a = 1n/n and b = 1/n. 
Official URL:  https://www.smartchair.jp/hp/PPBA2021 
Depositor:  MorneauGuérin, Frédéric 
Owner / Manager:  Frédéric MorneauGuérin 
Deposited:  29 Mar 2022 19:52 
Last Modified:  14 May 2023 23:21 
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