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IMA Journal of Mathematical Control and Information Advance Access originally published online on March 14, 2007
IMA Journal of Mathematical Control and Information 2008 25(1):75-84; doi:10.1093/imamci/dnm006
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Properties of the weighted logarithmic matrix norms

Guang-Da Hu{dagger}

Information Engineering School, University of Science and Technology, Beijing 100083, China

Mingzhu Liu{ddagger}

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

{dagger} Email: ghu{at}hit.edu.cn, ghuca{at}yahoo.ca

{ddagger} Email: mzliu{at}hit.edu.cn

Received on January 31, 2006; Accepted on December 9, 2006

In this paper, we are concerned with the properties of the weighted logarithmic matrix norms. A relation between the elliptic logarithmic matrix norm and the weighted logarithmic matrix norm is given. Based on Lyapunov equations, two weighted logarithmic matrix norms are constructed which are less than 1-logarithmic matrix norm and {infty}-logarithmic matrix norm, respectively. Then, an iterative scheme is presented to obtain the logarithmically {epsilon}-efficient matrix norm. Numerical examples are given to illustrate the results.

Keywords: the logarithmic matrix norm; the logarithmically {epsilon}-efficient norm; Lyapunov equation.


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