an:05275492
Zbl 1144.15018
Hu, Guang-Da; Liu, Mingzhu
Properties of the weighted logarithmic matrix norms
EN
IMA J. Math. Control Inf. 25, No. 1, 75-84 (2008).
00218705
2008
j
15A60 15A24 65F30
logarithmic matrix norm; logarithmically \(\varepsilon\)-efficient norm; Lyapunov equation; numerical examples
The properties of weighted logarithmic matrix norms are studied. The logarithmic norm of a matrix \(A\) is defined by the relation
\[
\mu[A] = \lim_{\Delta \to 0_+} \frac{\|I+ \Delta A\|- 1}{\Delta},
\]
for the matrix norm \(\|\cdot\|\) induced by a vector norm in \(\mathcal {R}^n\). The elliptic logarithmic norm, the logarithmically \(\varepsilon\)-efficient matrix norm, and the weighted logarithmic matrix norm are discussed. An equivalence relation between the elliptic logarithmic matrix norm and the weighted logarithmic matrix norm is presented. Based on the Lyapunov equation for the matrix \(A\) and a symmetric positive definite matrix \(H\), two weighted \(H\) logarithmic matrix norms are constructed which are less than the \(1\)-logarithmic norm and the \(\infty\)-logarithmic norm, respectively. An iterative scheme is presented to obtain the logarithmically \(\varepsilon\)-efficient matrix norm. Two numerical examples are presented.
V??clav Burjan (Praha)