A relation between the weighted logarithmic norm of a matrix and the Lyapunov equation
type: Item
instance of: scholarly article
title: A relation between the weighted logarithmic norm of a matrix and the Lyapunov equation
zbMATH Open document ID: 0963.15030
DOI: 10.1023/A:1022380116022
author: Guangda Hu
author: Guang-Di Hu
published in: BIT
publication date: 2001-02-06T00:00:00Z
review text: The weight \(H\) logarithmic norm \(\mu_H[A]\) of a matrix \(A\) is defined as \[ \mu_H[A] = \lim_{\Delta \to 0} \frac{{\|I + \Delta A \|}_H - 1} {\Delta}, \] where \(H\) is a symmetric matrix and the weight \(H\) norm of \(x\) and the weight \(H\) norm of \(A\) are \(\|x \|_H = \sqrt{x^T H x}\) and \[ \|A \|_H = \max_{x \in R^n, x \not= 0} \frac{\|Ax \|_H} {\|x \|_H}, \] respectively. For the case of a real stable matrix \(A\) the relationship between the maximal eigenvalue of the symmetric positive definite matrix \(H\) satisfying the Lyapunov equation and the weight \(H\) logarithmic norm \(\mu_H[A]\) is derived and it is shown for this case that \(\mu_H[A]\) has a negative value. Two examples for the case of \({3 \times 3}\) matrices are given.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1533025
zbMATH Keywords: weighted logarithmic norm
zbMATH Keywords: stable matrix
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: maximal eigenvalue
MaRDI profile type: Publication
full work available at URL: a:1022380116022
OpenAlex ID: W116296291
description: scientific article; zbMATH DE number 1533025